Click on Active Subscriptions under the product in question, you will see the list of your Parallels subscriptions.
Click on the subscription you would like to change payment method for.
Click on Payment Options, and you will be redirected to the page where you can manage the payment method.
You can also check this article: https://kb.parallels.com/en/123883
As for your query about changing computers, just log out from one computer (Parallels Toolbox icon > Cogwheel > Quit Parallels Toolbox)
Then activate on a second computer.
Let us know if you have any additional questions.
Wishing you a great week ahead!