But they might think I'm gay. Horse riding isn't the sort of thing they'd picture me doing.
okay. I'm guessing this is the cause of your doubt.
I can tell you from experience, sometimes the people you love, even the people you depend on aren't going to be as supportive of you as you might hope.
Present this to your parents. Don't be upset with them if they don't understand or support you. It'll hurt but you've already half prepared yourself for that possibility.
Next question is: are they willing to pay for lessons? If not then you have to go to plan B, which is you need to pay for lessons yourself with a part time job or some savings you may have (look into group lessons and buying them the most economically which can be in a group of five or ten lessons) or find a barn where they'll exchange riding lessons for work. You're young, presumably strong (if you're not, you will be once you start riding.)
I'm not sure of the laws in your state, whether or not you can ride without your parent's permission, but I imagine if you're willing to work to pay for lessons, your parents will support your commitment and sense of responsibility and at least permit you to pay for your own if they're not willing to pay for them.
What may help you is your parents seeing your determination and maturity. Being doubtful of your own sexuality is a bad place to start. I'm going to give you a piece of advice here and that is to quit asking if it's masculine to do anything.
All your life, you're going to be called to do things, you're going to be drawn to activities and interests, you will find you have may have born skills and natural talents to do - like cooking, like sewing, like caring for others (being a doctor). You have to go into them not caring about whether or not what you'd like to do, where you'd like to go or be fits "other people's" ideas of what you "should" be doing.
Your life is only yours to live. ONLY YOURS. You will live your life inside your body for as long as grace allows. One day you will die. Don't die wishing you had had the courage to persue your dreams. Let love and courage define you. Not other people.