Ultimately it is always going to be completely up to you who you date but I can understand why they are concerned if you have had a rocky relationship in the past.

All I can suggest is being completely honest - it is better to have the difficult, awkward discussion now rather than have your dad find out in a years time that you haven't been honest with him - trust me I have been there! It is much more difficult and awkward later on. It also then raises suspicion on their part when you feel you need to hide the relationship. They may start to question whether there is a reason you couldn't tell them and they shouldn't trust him.

As for building trust, all I can suggest is to give it some time. In time, if he sees that you are serious about your boyfriend and that he is a nice person who is good to you then normally trust will begin to form again. Dad's are particularly hard - they want to protect their daughters - but give it time.