Where is the pitot static drain located on a Piper PA-28?
This presentation gives a nice overview of various systems on the Piper Cherokee series. The buttons you describe are shown here:


(slide 9)

The full pitot-static tube including the drain is shown here:


(slide 11)

So the drain is incorporated in into the pitot-static tube itself.

The buttons are also acting as a drain themselves, for details see Alexander's answer.



Related:


What is the difference between a pitot tube and a pitot-static tube?