I just recently got my own horse (I've been trying to get one for 3 years) and what seemed to work out for me is showing my parents a bunch of different horses online (at lowers prices ranges) so your kinda "putting an idea" in their head. What really sold my parents was bring them with to look at saw horses, that way they could see there personalities. I hope it al works out for you!
I would NOT buy it behind your parents back. That is asking for trouble. Prove to your parents that you are responsible enough and that you understand and are willing to do the work to take care of it. Offer to pay for it yourself. If you buy it behind their backs then what happens when your horse gets sick and you can't pay for the care it needs? That would end very poorly.