Work really hard and show them your ready
It all depends on your experience in riding and how much you know, because you should not be getting your own horse if you no nothing about this responsibility. How I got my first horse is that I worked for it! I showed my parents that I can take care of my own horse.
It depends on your age, I got my first lease at 16 and I was working and paying for it myself, I was able to get her moved to the stable literally at the bottom of the road to my school and I was insured and responsible. It’s mainly showing them you are trustworthy and responsible enough to care for another life.
Tell your parents that you've been thinking about it since a long time, it is not a rush decision.
Tell them you think you will improve better with your own horse and you want to establish a real relationship with an animal, that it will bring you helpful values for your entire life, not just horse riding.
Show them also that you can combine horse and studies without difficulties (this is usually why they don't want you to get a horse), that it won't affect your grades.
Of course you need to think about the money so tell them that you can work to help them pay for it.
I hope i could help you :)
Show that you know what you are doing, and that you will know what to do when you get a horse. Make sure you have enough money as well. Take lessons for at least 2 years.