|I will probably NEVER buy another OEM version of an OS again, though. It's a pain. Change components or systems, and you can't use the software.
Similar things happen with certain software copyright checks. I find virtualization to be very handy for this actually as you always have a standard set of hardware presented to the guest OS. While I use the software involved perfectly legally and as intended, it makes things MUCH easier on me to do so on occasion.
To try to simplify things for the OP....
I do not believe it is technically legal to use OEM versions of the software in the way you are describing. However I don't have a EULA for the OEM version handy to be able to check this. Even if it is technically legal as you purchased a new computer, it is against the spirit of OEM software as it is intended for system builders to be able to install software on a custom built system, either for personal use, or the intended purpose, for selling as a single unit.
This is all seperate from whether it will technically work or not. It should work fine. I can't think of any instance where it has not I should say. The above only addresses the ethical issues of it all.
Technically, I can't recall if Parallels can run a 64 bit guest OS, so I will refer to the above and say no. You would want to get 32 bit. I believe VMWare Fusion may support 64 Bit guest OS if I remember right.