Coco looks like a sweetheart, and I love those blue eyes!
Have you considered making a "business" plan to help show your parents that you can make school and your horse work, and work well? Work up a plan that shows how you plan to use your time...how much time in school, how much time studying, how much time with the horses, etc. Set goals for yourself as far as grades, be realistic, but aim for an improvement over the past.
List the ways that having Coco help you grow and become a better person. That you are learning responsibility. That he relaxes you and that you study better after being with him. Explain that with your career goals, he is part of your education. Show that you are very serious about becoming a vet or physical therapist. Look into volunteering, job shadowing, college counselors, etc. Research colleges, and get your parents involved.
Remind them that colleges look for more than academic achievement. While grades are very important, they also want to see activities and community involvement. Showing Coco, taking him to nursing homes and/or children's hospitals and other activities will be fun AND will look good on college applications.
So many teenagers spend their free time just hanging and getting into trouble. You have something you're committed to. I commend you for this, and I hope your parents can see that Coco is helping you rather than hurting you.
Your parents may not understand now, but remember that they are doing the best they can to help you fulfill your potential. Show them in a way THEY can understand that Coco is helping you do exactly that. Prove to them that you are well on your way to being the person they hope you will be.
Good luck!