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!
I was stuck in this situation at one point and time but my parents didnt hang around. They took me and dropped me off and left me for 3-4 hours on weekdays and on weekends I was just stuck out from early morning to late at night. I would pack snakes and drinks and plan a whole day out there cause they wouldnt wait an hour and take me home. It was "okay if I take you I drop you off and then I will come get you whenever I have enough time".
Holy cow what spelling mistake. LoL SNACKS. LoL gee wiz....
I can understand not wanting to drive 40 minutes. My parents would have never agreed to that in the first place. That's an hour and twenty minutes drive time alone. I don't know what to say about getting them to drive you that far.