It's not you they don't trust, it's the other people you may come onto contact with and be influenced by that they don't trust. When you are a toddler they didn't trust that you wouldn't come into contact with the stairs or the edge of the coffee table. When you are 21 they won't trust that you will be safe while you are out drinking, when you are a teen they won't trust that you are mature enough to make the right decisions. That's what parenting is all about and I hope you will be the same way when you become one to protect your child.
The best way to deal with parents who don't trust you is to find a way to trust like showing a sense of responsibility. Let them know that you are now a grown up and that they should trust your judgement and decision.
I say tell them how you feel. They probably aren't ready to face the fact their "baby" is growing up. Ask them if there's anything you can do to gain their trust and work really hard to show how mature you can be. I totally sound like a psychiatrist/ therapist.... But still good luck!!! :D