If you feel invested in this girl, then I would take some time to negotiate the parents problem. Why don't you get to know them better by going to their house?

If you do that, I would take the opportunity to explain them that you have no problem in and getting to know the family, but a date is something personal in which pre-teen (or even mature people) have no place. If you speak to them directly, and in their home, they will be less inclined to dismiss you than if you remain an abstract concept for them. Don't be confrontational (that would be the worse), but show that you are serious, and that they can't easily brush you aside. This should make things much easier. Also, if they insist on this chaperone thing, then talk openly about what are their fears. They will be so embarrassed that you will have an advantage. In any case, negotiate your privacy with them.

At the end of the day, it's your girlfriend who needs to gain her independence, but if her parents see that you are serious and present, they will have a very hard time ignoring you. Also don't forget that they may be afraid of the unknown, but that if they get to know you they may be more relaxed