What happens, is that somehow you are lost as the owner. I am running Parallels from the MacBook harddrive, but the VMs are on an external USB drive. Whatever causes the issue, Parallels or Snow Leopard, your ownership is lost. In my case, the whole external drive was affected. I figured it out when Time Machine, also using that drive, said it couldn't write to the drive.
When I did a Get Info on the drive, and looked at the permissions, it said that me, the user, had read and write permissions, staff had read and write permissions, and everyone had read only permissions. All were grayed out. I clicked on the padlock, and unlocked it, but couldn't change any permissions as they remained grayed out.
I ejected the drive, and then remounted it. Everything was fine again. So, somehow, you become "everyone" and loose ownership! Scary.
If the VM's are on your harddrive, I would shutdown, reboot, and run Disk Utility / Repair Permissions on your drive.
Was it Parallels Desktop 5 for Mac? Was it Windows 7? Was it Snow Leopard?