Yes above can be done, but you would not be able to order any products, pizzas, etc through Alexa on the Echo under your account anymore. Parents have to buy into your idea with understanding you would not make any purchases on the Echo tied to their account. It would also make sense if you had another Alexa device that is on your account for your own use, and only use the one on Parents' account fot listeming to Prime music.. Getting a less expensive Dot for your account could solve this problem.
Keep in mind any changes to Alexa app on Echo would make changes on theirs, so if you use the Alexa app, only register it in your (new) devices account in your name, not theirs, There could be other pitfalls with this strategy, too, but it is not our job to think of them all; it's yours and your parents.