No.
Unless you have a multiple OS license (unlikely) you can only use your 1 key for 1 computer. The 32 or 64bit versions are not locked down to a specific key. So you can use your Win 7 upgrade key with both 32 or 64 bit versions. Just not on more than 1 computer.
So if you use your Windows 7 upgrade key on 1 computer, you will not be able to use it on the other computer. Also - you can use either 32 or 64bit version, although I reccomend the 64bit version as it supports more than 4GB of ram.