Re: I'm getting my own job soon, so my parents want me to be responsible for my own bill now
Based on my experience transferring the billing responsibility within my own family account, I would suggest that you give your parents your portion of the bill. If you are on a family plan sharing phone minutes and you create a new at&t account just for your number with your name on it, you will probably end up paying more money every month for voice because you would need to subscribe to an individual voice plan. Data and text messages could also become more expensive if you are subscribed to any grandfathered plans that are no longer offered.
On the other hand, you might rather pay more money instead of going though your parents.