How can I convince my parents to let me play small gigs?

Take your parents to one of these gigs where you are not playing so they can see it, it's sounds to me like they just don't understand it. Also tell them that you'll be safe because they can always be right there with you, watching you, driving you and taking you home. If that works over time they will give you the chance to do it on your own. Good luck and never give up on your music!