Are my parents the only ones this strict?

Are you the only daughter?
I don't think the problem has anything to do with trusting you. Maybe they don't trust HIM.
Its natural for parents to feel that way though. Especially if he is your first boyfriend, or if you are the only daughter.
You can try talking to them, and asking them why they act that way.
But if you don't want to confront your parents, then I suggest you have your Boyfriend show them he is trustworthy, and have your Boyfriend prove to them that he really cares about you, and respects you.
Time and effort is the only way he can win their trust.
Have you ever heard of courtship? Your boyfriend will have to use that method on your parents. you know, the old fashioned way.
Oh, and your parents aren't the only ones this strict. My parents are worst. I think.

find out why they don't like the guy
Either your parents are really prude, or they just don't like the guy

Talk to them and see why they don't like him
You can also have your parents talk to him

This really reminds me of my friend and his girlfriend. Do you have any friends that are couples? You could try going on double dates with him. Parents are really strict because they don't trust him not to make a mistake. Do you have family dinners? If you do you could try inviting him over for dinner frequently, so they can see what he is like.