Some questions for you to consider...how much do YOU think it costs to own a horse? How much do YOU think it's going to cost YOUR parents per month/year?
Okay, you have board, vet, farrier, worming. What about the unexpected vet bills? The unexpected farrier visits? The show costs if you choose to show? The cost of tack? The cost of show clothes? Training fees? Lesson fees? Gas for YOUR parents driving YOU to and from the barn to ride? Transportation fees to get your horse to and from shows or vet if vet cannot make a house call or horse requires to be moved?
Have you considered the sacrifice YOUR parents would have to make for YOU to have pleasure and enjoyment? I think it's pretty nice of your parents to even let you have lessons. I wasn't even able to take lessons growing up because it was so expensive for our family. I had the PRIVILAGE of being able to ride for about three years (and not consistently).