Dont listen to them, you can get peoples trust back, depending on who and what they are like, all i can say is suck up to them, but not to much as it can get quite annoying sometimes. Just show them that you really care about there son, because they will want someone who does. Just be nice and respectfull as that is what parents want.
Show remorse for whatever it was that made them lose respect for you,go out of your way to do the opposite, you need to do this on a continual basis as one offs dont count
You may never be able to gain their trust or respect again, but just continue to be true to yourself and respect yourself and others. Try to be honest without hurting others and just maybe if you improve yourself others will learn to forgive, trust and respect you again. If not you can be satisfied in knowing you are the best person you can be. Take care
Maybe do something very mature and do nice things for them
Do not act , be urself , u cannot act all the time
Just keep on being nice respectful and responsible. You will soon have a good connection with the parents. I don't think you should buy them huge gifts cause they may think your buying their friendship. Hope this helps! :)