Horses cost hundreds to thousands of dollars outright then cost several a hundred a month to board/feet/vet in a stable.

You shouldn’t be asking you parents to spend that much money on you, wait till you have a job.

First, convince your dad why you want a horse.
Tell him every thing you would do to keep it healthy and happy, like brushing it every day, feeding it twice a day everyday, raising your own money, picking the pen, buying food and tack, ect.

Do what I did and buy your own. My parents said no horse too, so I raised the money myself and bought my own horse!

Sit down and neatly and fairly work out on paper the cost of upkeep for the horse. Include everything. Then show how you will contribute to the care and wellbeing of the horse in detail.
By doing this you will show that you have done your research, know how to look after it and are willing to do everything you can.

Raise your own money. Your dad can’t say anything then because your using your own money. And since your mom already said yes then he shouldn’t have a problem with it.