Show your dedication first by volunteering at a local rescue barn or stabling facility. It is easier to convince your parents if you show them that you really want it and are willing to do whatever work is needed in order to participate in the sport.
well one of the major things is, are your parents not huying you a horse because they dont want to? or are you parents not buying you a horse because they arent in the financial situation to do so. it is pointless to beg for a horse and you parents say no because you dont have the money to afford and manage a horse. this is something you should discuss. i wasnt able to buy my horses until my dad got a promotion...before i wasnt in the financial sitiuations for a horse. so i would talk to them about the momey aspect of horses, as that is usually the main holdback and concern for people looking to buy horses.