Parallels does not delete or erase files. It will just remove the virtual machine from the Virtual Machine List window. It may also move files to the Trash. Files are not deleted until you manually empty the Trash so always check the Trash before emptying it.
The "Ignore ownership on this volume" option is only available on external disks (FireWire or USB or whatever).
Time Machine may or may not backup the virtual machine files, depending on your Parallels or Time Machine settings.
Check the permissions of the virtual machine files with the following commands:
The only part of the virtual machine that you really need is the .hdd file. The rest of it you can recreate by making a new virtual machine which you can then set it's Hard Disk 1 to the old .hdd file. So check that the .hdd is not corrupt by opening it with Parallels Mounter.
ls -ld ~/Documents/Parallels/*.pvm; ls -lRA ~/Documents/Parallels/*.pvm