How to show my parents I'm responsible?

How old are you?

Your parents just want to make sure you are safe and that you are not doing anything you shouldn't be. You can show them you are responsible by picking up more responsibilities around the house to start. Or try to find a few odd jobs like mowing lawns cleaning houses, taking a newspaper route etc.

Start using your time more responsibly. Don't spend all of your extra time hanging out with friends. Start looking for a career that you actually want to and start using this time pursuing that career. Or just ask your parents if there's any errands you could do for them.

That kind of trust comes with time and work. It's not just going to happen.