It gets worse. I decided to see if I could change any settings to improve performance. Started Parallels Desktop then clicked the gear icon in the small control panel window. The settings window opened, then I got a beach ball which after a few seconds stopped spinning. And at that point the whole Mac was frozen. Could not move the mouse pointer, or switch between programs with Cmd-Tab, or anything. Laptop fan started up, indicating that something was doing a lot of CPU processing. Eventually, after about 10 minutes more, the beachball started spinning again and I was able to move the mouse pointer. Cmd-Tab now brought up the row of icons for all running programs, but if I tried to select one to change focus to that program (or close one using Cmd-Q) nothing happened. After another 10 minutes or so, everything woke up and started operating pretty much normally. I made a few minor setting changes and quit Parallels. Restarted the Mac, since I did not trust it to be in a fully sane state.
It is clear that Parallels Desktop 12 has major problems in High Sierra, but if I spend my $50 and it is no better, I am out the money with no recourse.
I am going to see if I can use the 14 day free trial as an upgrade option.