I've never had sea horses but I did have a Dad that wasn't too happy about my fish hobby (okay, more like obsession)...so here's some advise:
- You're parents have it in their head this hobby is negative...telling them more negatives are just going to convince them more you don't need sea horses.
- Aquariums are better than furry pets: they don't make messes on the floor, chew up stuff, dog holes in the yard, sharpen claws on furniture...etc.
- Aquariums are a great way to make learning fun! Marine tanks are garanteed ways to learn some math & science in a way that is interesting to you.
- Aquariums teach RESPONSIBILITY! Parents absolutely love that word

I bet part of the reason that your parents are against you starting this tank is that they are worried you'll lose interest and they'll have to care for them. Maybe don't go sea horses right away...start small, see if they would let you get the sea horse environment ready (microalgae, clean up crew and such) and care for those things for awhile to prove that you are committed to what it takes to care for these guys properly.
- It's summertime, get out there and find a summer job of some sort...once again you're showing responsibility by working so YOU can be RESPONSIBLE for the cost of your new pets.
- Find an aquarium club in you're area and get one or both of your parents to go to a meeting with you, who knows maybe someone there has sea horses and can help you educate your parents on how great this they and this hobby can be.
Hope you can think of and add more to my list of positive influences an aquarium hobby can have on your life as a whole...and if all else fails you can always try the way my Mom would convince my Dad my tanks weren't such a bad thing for me to be interested in...she'd always say, "Would you rather our child be interested in dating?" He never agrued back on that one lol!
Good luck, just be patient, understanding and respectful with your parents about this issue and hopefully your parents will start to understand your interest in this hobby...who knows they might even learn to love the hobby themselves if educated about it prproperly...sea horses could be a cool & fun family bonding experience.