Google says there is no statute of limitations for this. Like others, I am surprised it was not part of your registration fees, but it has nothing to do with your income taxes not owed at the time.
Personally, I would do a little more research, but probably send off the check by the end of the day. If I were a long term record keeper, I would look at the registration from back then as confirmation.
As for the roommate, they were likely in violation of the registration rules at the time. They are even more likely to owe the taxes, which in many states go by physical presence, rather than registration per se.
I wonder if the state recently got a hold of the university's parking permit records and cross referenced with their own. Seems like a lot of resources for low return, like library fines or drug testing for welfare.