Does the activation key have to be purchased with the same e-mail as the account it is used in?

Hello,
Indeed, you may use an email address for the purchase and another one to register your Parallels Key.
To do so:

1. Login to Parallels Desktop Application with your Parallels Account.
2. Click on Parallels Desktop from the application menu bar on top of the screen; you may click on your Control Center window to see it. Select Account & License from the drop-down menu.
3. Click Enter a key.
4. Enter your license for the current version.
   If you are activating an upgrade copy, enter your key for the previous version. Click Activate.
   Kindly refer to this article for more information: https://kb.parallels.com/124225

Thank you