Just following up here. After some back-and-forth with Parallels support, I got an e-mail from a gentleman who must have seen my post on their forum (he referenced a screen shot that I don't believe I posted anywhere else or mailed to them directly) who told me to reinstall Parallels Tools. I was skeptical, as I had done this several times the day of the fail with no results, only this time it worked. I strongly suspect that they reset some permission or other, but that's speculation. So, I'm back up and running with Parallels 12 and OSX Yosemite 10.10.5. I keep getting "there's a new version" messages when I start Parallels but I'm going to let my blood pressure go back to normal before I try that again 
...and a final update: Today I updated my Mac to OSX 10.13 High Sierra and then updated Parallels to the latest version, and while it took a while for Parallels to get itself sorted, it all works. Thanks to everyone who offered input on this!
