OK, everyone.
I had the same problem.
What I did was run the terminal commands referenced earlier, start the Parallels Desktop.app, then ctrl-c the dispatcher in the terminal and run the same command again really quickly.
That launched the Parallels VM picker.
It works fine, but I do see this in the terminal:
WARNING: QApplication was not created in the main() thread.
/Library/LaunchAgents/com.parallels.vm.prl_pcproxy.plist: No such file or directory
QFileSystemWatcher::addPath: path is empty
QFileSystemWatcher: failed to add paths: /Library/Logs/DiagnosticReports
/Library/Filesystems/prlufs.fs//Support/prlufs.kext failed to load - (libkern/kext) not loadable (reason unspecified); check the system/kernel logs for errors or try kextutil(8).
the MacFUSE file system is not available (71)
/Library/Filesystems/prlufs.fs//Support/prlufs.kext failed to load - (libkern/kext) not loadable (reason unspecified); check the system/kernel logs for errors or try kextutil(8).
the MacFUSE file system is not available (71)
My VM (Windows 7) still seems to work fine at this point; I haven't hit any errors in the GUI.
Hope that helps.
-Bruce