There are two ways to find a license key on your Mac, both of them require further Parallels Support:
In Parallels Desktop interface
If you have Parallels Desktop installed and activated with the key you are looking for, and the version of Parallels Desktop is not less than 7, then you can find your product ID following the instructions from KB 111826.
When you locate your product ID following the instructions from the article above, please contact Parallels Customer Support to decode the product ID and get your Parallels Desktop 7/8/9/10 activation key.
In the Terminal
You can use this option to locate all Parallels Desktop keys used on your Mac for all Parallels Desktop versions, starting from version 5.
Start the Terminal (Finder > Applications > Utilities) and execute the following command:
prlsrvctl info --license
The output will give you the serial line with numeric codes in the following format:
The more activation keys you used on the Mac, the more codes you will see in the Terminal.
Copy the contents of the serial line and send to Parallels Customer Support.
For other ways to retrieve an activation key for Parallels Desktop for Mac visit KB 115424.
For information on retrieving license keys for Parallels Desktop for Mac versions 2.x or 3.0 please visit KB 4661.