I could only install Parallels 15 on Big Sur after removing a previous installation of VirtualBox. (It had kernel extensions that messed up the dependencies of Parallel's kernel extensions.) Even after getting Parallels running, its kernel extensions are still shadowed in /Library/StagedExtensions/Applications/Parallels Desktop.app, even though they successfully load. Parallels doesn't quit gracefully; I have to force-quit it.
I too noticed the bizarre bug of the "t", "y", and "2" keys not registering unless I create modifier-key-less shortcuts for them in the Parallels preferences. That's almost a usable workaround... in another bizarre twist, there's a short delay after typing shortcuts, during which no other key registers. E.g., if you type "tu" in rapid succession, you just get "t", but if you wait a little bit in between you get both characters. I hope this weird bug is easy to squash, because it's extremely annoying.