If you feel uncomfortable or under more pressure when you're with your parents, then don't take them with you, but ONLY IF it's for that reason.
From my Birmingham interview, I can tell you that many people brought their parents along with them (about half). No one judged on the basis of whether their parents came with them or not. The interviewers don't even know whether your parents are there, as they're in a separate room. Indeed, it even tells you on the tour reply slip that you should invite your parents in if they accompany you.
If you don't feel like you'll be under more pressure if your parents are there, then if they want to give you a lift up there, I'd go. It's so much more reliable than public transport and does not show that you are in any way less independent.