I'm no expert but I would guess that a program like Parallels puts a lot of software all over the place in OS X to be able to work. And once you upgrade, I would guess that some of that software is updated such that it doesn't work with old versions. If so, you can't downgrade by just dragging the P7 app back into "Applications".
I'm going to guess that probably the ONLY way would be if you backed up your whole system before you did the upgrade. If so, you can restore the backup to how things were then and that should restore P7.
However, I'm not even sure that will do the trick (if you can). I would guess that the nature of Parallels is to also put some software on the Windows partition or maybe update software within the virtual machine if you are not using a bootcamp partition. If so, I would guess THAT would have to be restored back to how it was before the upgrade too.
Maybe someone else will have a better answer for you. Good luck.
