Find the Previous System Folder, Get Info from the Finder (Command-i), change the owner of the folder to your own account in the Ownership & Permissions section, and then click the "Apply to enclosed items" button. You might have to enter your administrator password to do this, and it could take a while to go through it all. Normally it would be a fantastically bad idea to do this to a live system installation, but since this is an archive of all the old stuff that you're just going to harvest important files out of, it'll be fine. If any other files give you trouble after that, check the permissions on the file in that same part of the Get Info window.