I have Parallels Desktop 5 for Mac. Does the upgrade to Parallels Desktop 6 means that I have to reinstall Windows 7 and all the other programs?
No, your Windows VM and all the programs installed on it are stored on a virtual hard drive file. Upgrading Parallels won't affect the VM.
To extend @Kyle's answer, which is correct by the way, you have to consider the fact that after the upgrade, you might have to re-authorize Windows. I really don't know what Windows is thinking or what Parallels is doing, but I've seen two people who had to re-authorize their Windows 7 Home Premium inside a VM after that upgrade from Parallels 5 to 6.
This was not a problem and is generally performed "automatically", but just have your product key handy just in case.
I haven't personally upgraded to 6 yet (and probably won't). Paying 40 dollars a year to Parallels is more expensive than having Windows dedicated machines…
I had not trouble upgrading from 5 to 6 with Win 7 Enterprise.