OK, so there are a few things that caught my attention here.
First of all, you should never, ever, ever sell yourself short because you compare yourself to other people. If you love horses, then GO FOR IT. You should not compare yourself to other people!! Even a beginner horseperson or less experienced horseperson could go to college for horses. I mean, I could go to college for engineering if I wanted to and have never engineered in my life. You go to college to learn. If you feel as though this is something you are passionate about, then why not? If it is in your heart, that's awesome. You go to college to learn -- not to be perfect, not to get a career as a college freshman. You go to college so you can excel in what you love.
Who said that working with horses is not "making something of yourself?" I personally would love to be a doctor, psychologist, scientist... I've considered equine careers, but you know, I have time to think about it. The point is, you make something of yourself in whatever you want. We see these Olympic equestrians.. they have made something of themselves. We see everyday local horse vets... they too have made something of themselves. You will not waste your life with horses unless you expect to be successful with them easily. I do not have an equine career, but I've been working with horses for a decent while. I know that if you want to be successful, you'll have to push through it with blood (ex: if you pinch your finger in the girth buckle), sweat, and tears. Literally -- I've had all 3 as a rider, an equestrian, a horse woman. It happens, you'll get used to it, you just have to love it enough to get through it and pursue that final goal of _____ (whatever you expect in your equestrian career-- it could be one thing, or the list could go on endlessly).
Back to the parent thing. If your parents honestly love you, they should be happy with whatever you choose. My dad always wanted me to play basketball. Basketball, basketball, basketball. No, he would have much rather had me on the court, because that was his passion and basketball was so close to his heart. But you know what? Now, my dad loves my riding. He embraces it so much. He spends hours with me every week enjoying my passion, even though he can't ride to save his life. If your parents truly love you, they will be happy with whatever you choose. They might not be too thrilled at first, but when they see how happy you are, don't you think they'll be happy for you? I mean, every family is different, but why put yourself in a situation that you aren't happy?
Now, the phase thing. You can switch colleges, majors, whatever you want to do. But just keep in mind that as you are planning this out, you want to at least have an idea of what you want to do. So, God can change things in a heartbeat -- you could have an equestrian major one second and the next have this urge to become a doctor. Well, maybe not that drastic, but... anything's possible. My point is: you might change majors, you have that option as far as I'm concerned, but just know that before you commit, at least realize that you are passionate about it. I mean, you might have "fun," but if you aren't truly passionate about it, then eh, can you do it for the rest of your life?
Keep that in mind. From what it sounds like, and I CANNOT make any decisions for you, you sound like you should consider the equestrian career. You strike me that you seem very passionate about it, and that you are more worried about what other people will think. Not everyone relates to this, but I want to serve God above everything else in life. If this is a gift from God (meaning your passion for horses), do not swallow and hide that gift, but use it and raise it up to Him! I hope that you take these points and ideas into consideration, and follow your heart.
P.S. A lot of people make a living off horse careers as far as it seems... I mean, equine vets, for example. They seem to do fine. You know? It might not make as much money as a neurosurgeon, it might -- I don't have statistics. But you do what makes you happy, not for the money.