Tried this on Mac: chmod 644 ~/Parallels Things would go really weird, Guests could not stop or start but Parallels Desktop won't warn or report anything.
Once the permission of ~/Parallels was changed to 644 Windows could not shutdown, Windows Update would fails, After force quit of PD, Windows guest could not start. And PD would not correctly report this is cause by permission problem looks like PD would not check filesystem permissions.
One questions, why you didn't provide Parallels Desktop with the 755 permissions? Or it has been changed to 644 automatically or after any updates?
It's by default 644 on MacOS if I create a directory under $HOME directory and 777 is not secure under my environment.