Maybe one of the reason why your parents don't believe is lack of trust and communication between two side. Right now begin discuss with your parents about your every day life, that will improve your relationship become better in the future
If you are a teenage with no fixed income, it will very dangerous to leave your house to life outside by yourself, trust me. If you cannot prove your innocent, just be a good girl/boy.
Yeah they have. Almost often. running away from house is thing I always think about. But the next minute after that I realized, noone could ever deserve me than my parents did.