Bought a new computer--can you reuse serial number for new laptop

And to add to what Ann and Bill said, if you want to but can't deactivate it on the old computer:

Go to Adobe Customer Service and only ask to have your activations reset:
https://helpx.adobe.com/contact.html?rghtup=autoOpen will automatically open the chat window.

Type "Agent" into the text box to bypass the robot.

You may also be able to contact Adobe by phone:
https://helpx.adobe.com/contact/phone.html

Ann said most of it.  I'll add that you can use a serial number on two machines.  So if the old machine is not available or working, you can go forward with the new machine.  

If you don't already have it, you can get a fresh 2020 installation file here:  https://helpx.adobe.com/download-install/kb/premiere-elements-downloads.html?promoid=D4P81Y6G&mv=other

Your serial number is for Elements 2020.

If you want to run 2021 you need to buy the product first and get a new serial number.

If you don't want to buy 2021 download 2020 install and use your old serial number.

Make sure you deactivate 2020 on your old machine first.