I have been using Premiere Elements 11 on my MAC but now want to use it on my PC. I have Parallels running. Do I need separate PE11 for each, MAC and Windows?
Yes, it's a separate install for your Windows and for your Mac system -- assuming you plan to use it on both platforms.
BTW, Parallels is a TERRIBLE way to go if you plan to edit your video in a PC environment (IMHO). You're working in a Windows shell on the Mac system -- and that's just a horribly inefficient way to use your resources. Video editing is a very intense process. It's likely to give you many hours of headaches.
I very much recommend, as an alternative, Boot Camp, which lets you boot your Mac into a fully Mac or fully Windows environment. I use it on my Mac, and I can even do video editing in either Mac or Windows without so much as a hiccup. (It's also free.)