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Read everything you can find about horse care. Volunteer at a local stable. Take your parents to the stable. Stay devoted to horses.
You need to understand how and when to groom them, the best options for boarding them, how to muck out their stables, how to use and clean tack equipment and how and what to feed them.  You should also understand how to maintain a horse's health and what vaccinations it needs and when.  Study multiple sources.  Technical and medical advances revolutionize the horse care industry periodically. Learn what plants are poisonous to horses and how to avoid horse ailments. Let the stable owner know that you plan on owning a horse.  Try to do every job you can.  You'll gain valuable hands-on experience caring for horses while learning from an expert.  Work often, the more experience you have, the more comfortable you'll be taking care of your own horse. Ask the stable owner any questions you may have.  He'll likely have great insight. Show them everything you've learned.  Let them see you perform daily tasks related to horse care. Have them speak to the stable owner.  A good recommendation from him could ease your parents' concerns. Show your parents your dream to own a horse is more than a passing thought.  Keep working with horses as much as possible.  Be consistent.  It may take time before your parents understand that you truly want to own a horse. Bring your work to your parents' attention.  Remind them of your goal to own a horse often.  Say things like: "Today I mucked the stalls for all the horses in the stable.  I can't wait to clean up after my own horse." or "I helped hold a horse still while it got vaccinated today.  Did you know horses need to be vaccinated for tetanus every year?"