The rule doesn't have to do with your age, but rather with your driving ability. Since you do not have an actual licence yet (you have a learning permit, meaning you are allowed to learn, meaning you do not yet know how), you need someone who does sitting beside you to guide you and pretty much be responsible for whatever happens when you're behind the wheel.
The only thing you have going for you over a "Junior" driver is you do not have an age restriction over when you can get your complete license. They might want you to wait for a little bit, maybe 3 months, sometimes, though.
If you're finding the 21 year old in the car bothersome (not necessarily an ego thing, could be your schedules don't match or something), then you probably need to finish learning ASAP and get your full licence.
Any exceptions to the rule will be covered in your State's manual. They are frequently published online, so you can do a search.