Move Parallels to new Mac using ONLY the UPDATE key

I've been a Parallels user for about 5 years, using it on the same Macbook Pro all this time.
I don't recall which version of Parallels I started with, but I have been purchasing UPGRADES over the years and am currently using v10.
I just bought a new Macbook Pro and want to install Parallels on the new machine (and NOT keep it active on the old machine - going to sell it)

The ONLY license key for Parallels I can put my hands on at this time is the upgrade key for v10.

1) At http://kb.parallels.com/en/116527 I'm told that I first need to uninstall v10 from the old machine. Ok.
2) From that link I get referred to this article next http://kb.parallels.com/en/122653 Ok.
3) Now v10 is removed from the old machine. Ok.

So my question is this:
I now need to get the Parallels v10 dmg and get it installed using ONLY my UPDATE key. How do I do this?