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Write down everything you'll need to purchase and maintain your horse.  Include every responsibility you'll have as a result of owning a horse.  Include what expectations you'll have of your parents. Include what expectations your parents should have of you. Look for horses that will meet your personal needs.  Find the best prices on those horses in your area.  Avoid free horses, as they often have medical or behavioral issues. Learn the history of horses you may buy to guarantee their quality. Find out how much it will cost yearly to keep your horse.  Include equipment and medical costs.   Leave room for the unknown.  Include possible setbacks in your budget. Give them a plan for storing your horse. You can cut into the cost of showing your horse by learning to braid professionally for other contestants at shows. Decide how much of your own money you can spend on your horse.  Prove that you'll take horse ownership seriously by putting your own cash towards it.

Summary:
Make a proposal. Shop around for horses. Figure stable and grooming expenses. Determine your contribution.