Honestly, they actually don't sound too bad. Hell, my parents freaked out when they found out I had a guy friend, (not a boyfriend, a guy friend) and I'm sixteen. If I were to talk about cute guys on TV they would probably shoot me. If you're thirteen, I'm guessing you're only in 7th-8th grade? It's best to wait at least until high school because although you may feel like you're ready, you're probably not emotionally ready for actual dating, (as in going out for dinner, etc.) I've never really been able to understand how people under 16 or 17 can date. Would your parents have to drop you off at the restaurant? Sorry I'm sure you're dead sick of hearing people tell you the same Sh1t over and over...
If you really want to convince them, the best thing to do is to act really, really mature. Do your homework as soon as you get home from school, don't complain about chores, try not to get into any 'he/she started it' arguments with your siblings, etc. Your parents are always going to see you as their little baby, but maybe if you act more adult, they'll realize you're growing up. And if they still say no, don't argue with them. Whining about their decision will only exacerbate the dilemma. Besides, you catch more flies with honey than with vinegar.