I FOUND THE ANSWER:
The hang up is because Windows is showing an "Untrusted Publisher" alert for the Parallels Tools drivers, because they are not properly signed. The automated upgrade/install program can't detect the error, apparently, so it can't tell you there is a problem. It just sits.
SOLUTION: When the PT install stalls, switch to the installer's Manual mode by clicking the lock icon in the installer window (next to the large-type heading). When you see the Windows Desktop and the red warning dialog, hit "Install this driver anyway." You will have to do this several times, once for each unsigned driver. Do not restart the Windows guest OS and ignore and close any other dialogs, like "Could not find driver" or "please restart" -- you have to wait for the installer to stop working.
PD People: You can fix this by using properly signed drivers!
-paul