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Don't be pushy.  Deciding whether or not to buy a horse is a big decision.  Your parents will likely need time to think it over.  Show them you can be patient. Take the extra time to prepare yourself and continue to study. Do your best in school.  Finish all your chores. Stay out of trouble. Show your parents you can handle responsibility and are ready for more. Continue working with horses. Gain all the experience you can. Show your parents how passionate you are about working with horses. Even if your parents end up saying no, thank them for their consideration.  If you handle the disappointment well, they'll be more likely to change their mind later.  Don't blame them.  Say "I completely respect your decision.  I realize owning a horse is a huge commitment.  I'm still passionate about horses and I hope someday I can prove to you that I do deserve the honor of owning one." Understand that they may not currently have the means to support a horse. Don't give up.  Keep trying to convince them that you can handle horse ownership.
Give your parents time to consider. Follow through on other responsibilities. Continue to show commitment to horses. Act mature.