Learn to apologize. The first thing that you can do to get your trust back is to apologize for your mistakes. There are things that have let your parents down and that is the reason why you have lost the trust, so in order to gain it back, say sorry to them for the mistakes that you have made. That will really help your parents understand how sorry you are and they will definitely try to forgive you. However, time is the best healer, and they will be fine in a few days, weeks, months or years.
Since you have messed things up badly, you need to do something in order to make it up to your parents. Therefore, be reasonable. Be someone they think has changed. They should know that you are not the only person making the mistakes. Moreover, you should make them realize that the mistakes you made were a mistake and you are apologetic towards them. Therefore, always make sure that your acts define your intentions.
If your parents have punished you, simply accept that punishment. Some ground their children, some stop talking to them while others even stop them from doing certain things. Hence, whatever the punishment may be, you just have to be sure that you accept that punishment. Even though it hurts a lot to accept things as they are, but you have to do it in order to regain your parents’ trust. Therefore, always accept whatever the punishment may be.
Respect is the key towards everything. Respect comes before trust. Therefore, when you start respecting them, they will start trusting you automatically. Whenever you think you are talking to them rudely or responding to them in a way which is not appropriate, avoid talking any further.