Most parents are worried that there kids are just going through a pony stage, show her that this is what you really want to do and no matter what, stick with it. I had to wait 10 years of consistent riding and showing for my parents to finally get a horse for me. Dedication and hard work!!!
Make sure to save some money and have good grades in your classes. Show her you are disciplined and ready for some real responsibility. Approach her with good arguments and state your point without forcing it on her. Eventually she will come around.