How do I convince my parents to let me ride more? The main reason they wont let me ride is because my barn is far away and takes up a lot of time.

Find a way to go to the barn on your own and prove them it won't interfere with your school work and that the journey is safe.
maybe find a closed barn if you can, if not prove to them how committed you are to the sport that way they know there not wasting their time
Find a closer barn or show that you are really committed by helping out trainers, ferrier etc. Or find someone from your barn that you can carpool with.
You could find a closer barn. Or if you can drive you could drive yourself. Or make some kind of deal where they take you one extra day a week if some chores or something.
Explain to them how much it means to you and why you want to do
I sometimes have the same problem. Let them know how much it means to you. Or offer to do something in trade for riding more often
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