Your parents have 3 children (that includes yourself), correct?
I'd almost think that the price of gas might have something to do with it too.
As a courtesy, and to show your parents that you really, really want to go, you could take up some odd jobs around the neighborhood; babysit, mow lawns, paint, etc. That money can go directly into a jar that will pay for gas to and from the barn. Anything left over can go directly towards the care of your horse.
Make sure you do your own chores though, too! Proving that you're a responsible kid can go a lot farther than pleading with them!