Some parents just don't like horses and don't want their kids involved with them. My parents are the perfect example. I was horse crazy early on, I drew them, had the my little pony's, then moved on to Breyers, read all the horse books... you name it. But my parents didn't have any inclination towards horses. My aunt owned a stable, my cousins rode... it wasn't like we were on an island with no horses. But my parents remained steadfast and there was never a horse or lesson. They felt it was a phase I'd grow out of.
I never grew out of the phase. But once I went to college I couldn't afford it, then after college my job required lots of travel... fast fwd to a year ago. I finally have the money and time for a horse. I buy one and the parents still make it very clear they disapprove! Good grief, I've been married 8yrs, lived on my own longer, and completely independent of them yet they still felt strongly enough to bring it up. Come to find out after all this time it was due to my dad falling off a pony as a kid and he never got over it... didn't want me to get hurt.
So money aside, parents have their reasons and you may not be privy to them. But if it's really what you want you will eventually find a way to make it happen.