How can I convince my parents to let me try out martial arts (like aikido)?

I want to learn self-defense/martial arts, but I have health issues. I can't exercise a lot or I strain muscles, and I've landed up in the emergency room because of it before (when I tried doing more than 1 hour of DDR, a dance game). I want to learn martial arts so I can defend myself if I ever needed to.

I would tell my parents that every teenage girl should have self-defense skills, but they would say I have a phone in case of emergencies, and that I shouldn't be out late anyway (which I understand, but I'm a college student and my some mandatory classes end late).

Advice?
 
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