check your key bindings in the general Parallels settings (not the VM settings). Copy-Paste on the Mac is cmd-c / cmd-v. On the Windows side, it's ctrl-c / ctrl-v.
If you don't have the Parallels tools installed OR if you're not loading the prl_cc.exe within Windows OR if you've opted to isolate your VM (that's set in the VM's settings, not the general Parallels settings), copy-paste between Mac and Windows won't work.