I didint ask I waited till my parents realized I needed a horse to be ble yo progress in my riding. For me it had to be something that I needed and since my parents made the decision they are very happy about it( my trainer also convinced them)
I think that more than "convince" them, it would be about showing them that you are ready and responsable enough to care about the horse.
I think that you could make them a budget, to help them see how much it would cost them, and if they can afford it or not.
You could also make a sorte of list of how it would be good for you to have a horse (it would help you be more responsable, it would force you to spend a lot of time outside instead of being on social media, that sorte of things), and to finish with, you could also make a list of things that you agree to do in return (for example, help more with the chores or whatever could help them).
Don't convince let him decide its how I got my horse o didn't try to convince him by doing things I my dad decided when I was ready