Pieday 1, 0 bytes

Pieday does this by default when no input is entered

Pascal (FPC), 263 229 bytes

uses sysutils,RegExpr;var x:array[0..2]of string=('.*:03:14','.* 03:14.*','14-3.*');y:array[0..2]of string=('Second','Minute','Day');i:word;begin
for i:=0to 2do if ExecRegExpr(x[i],DateTimeToStr(Now))then writeln('Pi ',y[i]);end.

Try it online!   _263b

_{Might be shortened by using if..then..elseif.. instead of complicated arrays & loops.}

Pascal, 226 Bytes

See also FreePascal. This program requires a processor complying to ISO standard 10206 “Extended Pascal”.

program p(output);var t:timeStamp;begin getTimeStamp(t);with t do write(subStr('Pi Day   ',1,9*ord(month=3))+subStr('Pi Minute',1,9*ord(hour in[3,15]))+subStr('Pi Second',1,9*ord(minute=3)):9*ord(14 in[day,minute,second]))end.

Ungolfed:

program piDMS(output);
    var
        ts: timeStamp;
    begin
        getTimeStamp(ts);
        { `with t do ` is two characters shorter than six times `t.` }
        with ts do
        begin
            writeLn(
                    { subStr(string, firstCharacterIndex, length) }
                    subStr('Pi Day   ', 1, 9 * ord(month = 3)) +
                    subStr('Pi Minute', 1, 9 * ord(hour in [3,15])) + 
                    subStr('Pi Second', 1, 9 * ord(minute = 3))
                    : 9 * ord(14 in [day, minute, second])
                );
        end;
    end.

216 characters if

• the implementation-defined value of maxChar has an ordinal value ≥ 59 and
• neither chr(3) or chr(14) of the implementation-defined set of char values indicate an end of line (Pascal does not permit multiline strings).

program p(output);var t:timeStamp;begin getTimeStamp(t);with t do year:=index(chr(month)+chr(day)+chr(hour mod 12)+chr(minute)+chr(second),'');write(subStr('Pi SecondPi Minute         Pi Day   ',37-9*t.year):9)end.

NB: The string literal '' is chr(3) + chr(14).

Japt, 48 47 45 bytes

:\ I can definitely do better than this! Outputs undefined if it's not a Pi date/time. Second line include a trailing space.

`DÃÕsMÔSSeÖ`qÅæÏgKs6 f"%d+" ã2)e#;ìF
©i"Pi 

Test it, or test:

• Pi Day
• Pi Minute
• Pi Second

`...`qÅæÏgKs6 f"%d+" ã2)e#;ìF\n©i"Pi 
`...`                                     :Compressed string "DaysssMinutesSecond"
     q                                    :Split on
      Å                                   :  "s"
       æ                                  :Get first element to return true
        Ï                                 :When its index is passed through the following function
         g                                :  Index into
          K                               :    Date
           s6                             :    To locale string (M/D/Y, h:m:s A/PM)
              f                           :    Match
               "%d+"                      :      RegEx /\d+/
                     ã2                   :    Sub-arrays of length 2
                       )                  :  End indexing
                        e                 :  Is equal to
                         #;               :    59
                           ì              :    Converted to digit array in base
                            F             :      15
                             \n           :Assign to variable U
                               ©          :Logical AND with
                                i"Pi      :  U prepended with "Pi "

Zsh*, 93 88 83 _(98-15) bytes

case `date` in *Mar\ 14*)t=Day;;*\ 03:14*)t=Minute;;*03:14\ *)t=Second;esac
<<<Pi\ ${t?No Pi Time}

Try it online! (98-15 bytes)
 _{103-15 bytes  93 bytes  107-15 bytes, using associative array}

Used extendedglob for matching the case patterns. If $t wasn't set (${t?}), then it prints "No Pi Time" to stderr.

05AB1e, score: 31 (46 bytes - 15 bonus)

žfžeža12%žbDžc)2ôJƵÖk”€º‚Ž™Ù…Þ”#…Pi ìć…No ìªsè

Try it online or try it online with emulated datetime.

Adding the No Pi Time bonus costed 8 bytes, so it's worth the -15. Here it is without the bonus:
žfžeža12%žbDžc)2ôJƵÖk”‚Ž™Ù…Þ ”#„Pirèðý - Try it online or try it online with emulated datetime.

Explanation:

žf               # Push the current month
  že             # Push the current day
ža               # Push the current hours (24-hours format)
  12%            # Modulo-12 to make it a 12-hours format
     žb          # Push the current minutes
D                # Duplicate the current minutes
 žc              # Push the current seconds
)                # Wrap all values on the stack into a list
 2ô              # Split it into pairs
   J             # Join each pair together
    ƵÖ           # Push compressed integer 314
      k          # Get the first 0-based index of 314 in this list (or -1 if none are)
”€º‚Ž™Ù…Þ”       # Push dictionary string "Time Day Minute Second"
          #      # Split it on spaces: ["Time","Day","Minute","Second"]
           …Pi ì # Prepend "Pi " in front of each:
                 #  ["Pi Time","Pi Day","Pi Minute","Pi Second"]
ć                # Extract head; pop and push remainder-list and first item separately
 …No ì           # Prepend "No " in front of it: "No Pi Time"
      ª          # Append it to the remainder-list:
                 #  ["Pi Day","Pi Minute","Pi Second","No Pi Time"]
s                # Swap so the earlier index of 314 is at the top
 è               # Use it to index into the list of strings,
                 # where -1 would get the last item
                 # (after which the result is output implicitly)

See this 05AB1E tip of mine (sections How to use the dictionary? and How to compress large integers?) to understand why ”€º‚Ž™Ù…Þ” is "Time Day Minute Second" and ƵÖ is 314.

Go, 211 - 15 = 196 bytes

import."time"
func f(){N,s:=Now(),"Pi "
M:=N.Minute()
if _,m,d:=N.Date();m==3&&d==14{s+="Day"}else if N.Hour()%12==3&&M==14{s+="Minute"}else if M==3&&N.Second()==14{s+="Second"}else{s="No "+s+"Time"}
println(s)}

Attempt This Online!

Prints to STDERR.

CJam, 41 bytes

et[3E]#"
Pi Day

Pi Minute
Pi Second
"N/=

Test it here. Alternatively use this link to stub the result of et for easier testing.

Explanation

et   e# Get the current datetime as an array with the following elements:
     e#   - Year
     e#   - Month
     e#   - Day
     e#   - Hour
     e#   - Minute
     e#   - Second
     e#   - Millisecond
     e#   - Weekday
     e#   - Tickrate or something.
[3E] e# Push the array [3 14].
#    e# Find the position of this subarray in the current datetime array. Let's see
     e# what results we can get:
     e#   - Year 3 is in the past and there is no 14th month, so we can't get 0.
     e#   - Pi day will give result 1.
     e#   - Day 3, hour 14 would give 2.
     e#   - Pi minute will give result 3.
     e#   - Pi second will give result 4.
     e#   - Second 3, millisecond 14 would give 5.
     e#   - Weekday and tickrate won't be 14, so we'll never get 6 or 7.
     e#   - If [3 14] isn't found at all we get -1.
"\Pi Day\\Pi Minute\Pi Second\"
     e# Push this string (with linefeeds instead of backslashes.
N/   e# Split into lines.
=    e# Select the corresponding element. The non-existent "pi hour" and "pi millisecond"
     e# would map to empty strings as well as the -1.

PHP, 85 - 15 = 70 bytes

<?=['No Pi Time','Pi Day','Pi Minute','Pi Second'][strpos(@date(Ymdhi_is),'0314')/4];

The main trick use here is the Ymdhi_is date format, at the time of writing, date('Ymdhi_is') returns 201501030258_5828.

• md, hi and is are the values who will be replaced by 0314 if it's Pi-something. Note that all those strings will be always be replaced by a 4-character long string.
• They are put in that specific order since strpos will stop searching at the first occurrence of the needle, so we put them in the order of priority.
• A separator between hi and is is necessary because we don't want strpos to match a value that would overlap both (thanks to primo for saving bytes here).
• The needle is 0314 because 314 would wrongly match 10:31:42 as Pi-Second.

The Y part is the trickiest. We need a prefix to to offset the first occurrence of Pi-something, allowing us to distinguish strpos's return values between false (not found, Pi-nothing) and 0 (found at index 0, Pi-day).

And we want this prefix to be either 4 or 5-character long, since we are planning to divide strpos's return value by 4.

Y is 4-character long, but:

• it will be 5-character long someday, and this will break the program (think about year 10314): the PHP documentation says that Y will be replaced by 4 digits, but it's not true.
• if you come back in time, at year 314, it will break the program.

Since PHP didn't exist in year 314, and will likely not exist anymore in year 10314, I guess these bugs can be safely ignored.

Note that 0314 can overlap Ymd since:

• Ymmd configuration: there's no 31st month.
• YYmm configuration: there's no 14th month.
• YYYm configuration: there are less than 40 months.

Also, there's a version without the bugs related to the year, which is 86 - 15 = 71 bytes:

<?=['No Pi Time','Pi Day','Pi Minute','Pi Second'][strpos(@date(D_mdhi_is),'0314')/4];

Perl, 80 - 15 = 65 bytes

print'No 'x localtime!~/(ar | 15:|03:)14/,'Pi ',(Time,Day,Minute,Second)["@-"/4]

Take 2, parsing the string representation of localtime. At present, this looks something like this:

Sun Jan  3 15:14:15 2016

The position of the matched string is used to determine the correct Pi Time.

Perl, 100 bytes

@t=localtime;$t[2]%=12;3-/3/^{@t[$_,$_+1]}->{14}||exit!print'Pi ',(Second,Minute,_,Day)[$_]for 3,1,0

localtime returns the months zero indexed, hence the need for 3-/3/.

TI-BASIC, 124 bytes

Thanks to FlagAsSpam for shaving a few bytes.

"→Str1
getTime
If min({3,14}={Ans(2),Ans(3
"Pi Second→Str1
getTime
If Ans(2)=14 and max(Ans(1)={3,14
"Pi Minute→Str1
getDate
If min({3,14}={Ans(2),Ans(3)
"Pi Day→Str1
Str1

AppleScript, 202 190 187 183 181 bytes

Hey, this isn't so bad after all.

set n to current date
set b to n's time string
if n's date string contains"March 14"
log"Pi Day"
else if b contains"3:14:"
log"Pi Minute"
else if b contains"3:14"
log"Pi Second"
end

I actually found a use for AppleScript's method calling. Go figure. Nope. Just turns out that I'm an idiot. Setting a variable is shorter.

(for those wondering, current date command returns a date-type with the contents "Saturday, January 2, 2016 at 2:46:01 PM" or the like)

Javascript (ES6), 114 112 - 15 = 97 bytes

x=>['Pi Day','Pi Minute','Pi Second'].find((x,i)=>[/ar 14/,/(03|15):14:/,/03:14/][i].test(Date()))||'No Pi Time'

Ungolfed:

x=>
['Pi Day', 'Pi Minute', 'Pi Second']  // array of outputs
.find(                                // find first element in the array
    (x, i)=>                          // which returns truthy for this function
    [/ar 14/, /(03|15):14:/, /03:14/] // array of regex patterns
    [i]                               // get corresponding regex based on index
    .test(Date())                     // test it against current date, date is automatically cast to string
) || 'No Pi Time'                     // if no result, then return "No Pi Time"

Thanks for -2 bytes @edc65

Japt, 78 - 15 = 63 bytes

D=Ð)g ¥3©Df ¥E?"Pi Day":Dd %C¥3©Dc ¥E?`Pi M©e`:Dc ¥3©Db ¥E?`Pi SeÖ:No Pi Te

Pretty straightforward - just checks the date for every case.

Explanation:

• D=Ð)g get the current date (Ð), store it in the variable D and get the month (g). Why store it in the variable, if it's already one-letter? Because since then you can dress any part of date with Da, where a is the function, returning year, month, date, etc. But otherwise you'd have to do Ð a, see the space.

• ¥3 is ==3, checking if the month is March.

• © is &&, i.e. "and".
• Df is the day of the month.
• E is 14
• ?...:... - typical sets of ternary operators
• Dd %C the reminder of dividing the hour (Dd) by 12 (C)
• Dc is the minutes
• Db are seconds

Try it online!

To emulate Pi Day:

D=Ð"3/14/11 11:11:11";
Dg ¥2©Df ¥E?"Pi Day":Dd %C¥3©Dc ¥E?`Pi M©e`:Dc ¥3©Db ¥E?`Pi SeÖ:No Pi Te

To emulate Pi Minute:

D=Ð"11/11/11 3:14:11";
Dg ¥2©Df ¥E?"Pi Day":Dd %C¥3©Dc ¥E?`Pi M©e`:Dc ¥3©Db ¥E?`Pi SeÖ:No Pi Te

To emulate Pi Second:

D=Ð"11/11/11 00:3:14";
Dg ¥2©Df ¥E?"Pi Day":Dd %C¥3©Dc ¥E?`Pi M©e`:Dc ¥3©Db ¥E?`Pi SeÖ:No Pi Te

Python 2, 219 186 183 Bytes (198-15)

_{I tried}

Ungolfed:

from datetime import datetime

now = datetime.now()
output = ['Pi Day', 'Pi Minute', 'Pi Second', 'No Pi Time']

if now.month == 3 and now.day == 14:
    print output[0]
elif now.hour == 2 and now.minute == 13:
    print output[1]
elif now.minute = 2 and now.second == 13:
    print output[2]
else:
    print output[3]

Golfed:

from datetime import *
n=datetime.now()
a=n.minute
if n.month==3and n.day==14:print'Pi Day'
elif n.hour==2and a==13:print'Pi Minute'
elif a==2and n.second==13:print'Pi Second'
else:print'No Pi Time'

Python 3, 179 bytes

import functools as F,datetime as D
T,G=D.datetime.now(),getattr
F.reduce(lambda i,j:print("Pi "+i.title())if G(T,i)/G(T,j)==3/14else j,"month day hour minute second".split(" "))

Python 3, 137 - 15 = 122 bytes

import time
T=list(time.localtime())
T[3]%=12
print({1:'Pi Day',3:'Pi Minute',4:'Pi Second'}.get(bytes(T[1:6]).find(b'\x03\x0e'),'No Pi Time'))

The 12-hour requirement was unfortunate, as this would've been 118-15=103 bytes without it:

import time
print({1:'Pi Day',3:'Pi Minute',4:'Pi Second'}.get(bytes(time.localtime()[1:6]).find(b'\x03\x0e'),'No Pi Time'))

Ruby, 125 124 chars

i=[*[(t=Time.new).month,t.day,t.hour,t.min,t.sec].each_cons(2)].index [3,14];i&&$><<['Pi Day','','Pi Minute','Pi Second'][i]

Alas, the cleverer %i[month day hour min sec].map{|x|Time.new.send x} is longer.

The key here is the use of each_cons to avoid repetition (see the last few lines of the explanation below).

i=                          # send i (index) to...
[*                          # convert to array (splat)...
 [
  (t=Time.new).month,       # the current month...
  t.day,t.hour,t.min,t.sec  # etc... (duh)
 ]
 .each_cons(2)              # each consecutive two elements
]                           # [[month, day], [day, hour], [hour, min], etc]
.index [3,14];              # first occurrence of [3, 14]
i&&                         # shorthand for "if i"...
$><<                        # output...
[
 'Pi Day',                  # [month=3, day=14] is Pi Day
 '',                        # [day=3, hour=14] isn't anything
 'Pi Minute',               # [hour=3, min=14] is Pi Minute
 'Pi Second'                # [min=3, sec=14] is Pi Second
][i]                        # index by index (obviously)