I have a girlfriend and I really like her, but her parents are psychotically protective to the point of not even letting us hang out. What can I do to gain their trust?

It's commendable you want to gain their trust rather than sneaking around. Here are some suggestions:
When you call your girlfriend's house, speak politely to whomever answers: "Hi, it's John, may I please speak with Mary?" Good manners always have a positive effect. Try to dress neatly without compromising your own style choices when you visit their home. Inquire of them what would make them more comfortable -- would they consider letting their daughter go somewhere with you for 1 hour (at first) and when you have her back home AT THE RIGHT TIME, maybe they will eventually extend that deadline. What if the two of you are part of a group that goes to an activity together, would they accept that? This is a slow process, but building trust is never easy. Maybe until they get to know you better they would allow you to be at their home, say to "study" or watch a movie? Find ways to let them know you are a good guy - mention any good grades you might have, or how many hours a week you work, or anything that shows you have a sense of right and wrong, a good work ethic, plans for the future you are working toward, respect for women. Show an interest in her family. Keep your language clean too. Unfortunately, to win their trust you will have to be around them. There is no substitute for that. Good luck - again I am impressed you are willing to try. She must be special if you are willing to jump through a bunch of hoops!

Response to BeverlyofHills: If having manners is "fake" then I agree it isn't helpful. What a pity manners are considered "fake"!

They are either freaks or think you're a bad influence on their daughter. Find out which (I'm assuming the girlfriend could assist here). If you're a bad influence, um, decide whether you really want to screw up her life or not. Otherwise, find out what kind of eccentrics they are and play to their nonsense.

Seem more friendly, introduce your parents to them, anything to make them relise that your a good guy.

What does psychotically mean?

Maybe nothing. Perhaps you can encourage her to see a counselor so she can learn how to break those chains.