Is my Parallels installation configured just for the computer it has been installed on or can I install it on a different computer. I really want to move my entire parallels set up from my MacBook to a new MacMini.
You should be able to uninstall Parallels on one machine and install it on another. To uninstall: http://download.parallels.com/desktop/v4/docs/en/Parallels_Desktop_Users_Guide/23646.htm To get the key in case you forgot it: http://kb.parallels.com/en/5680
But I can first install it on the other computer, right? I want to be sure it works and that I get my VM over to the new machine prior to deleting everything on the MacBook.
I do not think it would be possible to have the same license key on two different machines at the same time. However, there is an easy way out of this, you can get a trial key here: http://www.parallels.com/download/desktop/ It is 14 days, so you have a plenty of time to check that everything is working properly.