What I found easy: copy to an external hard disk on the old PC and copy from the external hard disk on the new PC. The nice thing: you have a backup without doing anything extra.

Clearing the hard drive is more difficult. The definite solution is to use DBAN to totally erase it, then give it to your parents, say the hard disk is totally empty but Linux is free to install. Or add the system recovery media to what they gave you and instruct them to boot from it to bring this old PC back to factory conditions.
But I can imagine you're envisioning giving a somewhat less clear system than a totally clear system. You didn't really tell your intentions.

Kees