If you're under 18 and/or your parents are paying for any of the cost of the care, then there really is no "convincing" to be done IMHO. It's their money, so regardless of if they can afford it or not, it's their money regardless. It's not just "convincing parents to get you a horse", it really comes down to convincing them to spend X amount of their hard earned money each month, as well as potentially shelling out more money for emergencies and such. And that's where I think it's ultimately wrong to try and "convince" them to do this.
IMHO, if parents offer to pitch in, that's great, but if you feel the need to "convince" parents to spend money each month (plus unplanned emergency bills that can arise), then I personally don't think that's really right to ask of them.