Maybe you should show your mom that you are mature enough to handle baby sitting. A lot can happen. I know this because I am a parent. Maybe you use your phone too much or you put off your responsibilities. I think if you really want to know what you can do to prove to your mom that you actually are ready, (and, no offense), and not just think you are. I'm not saying that you aren't mature enough to babysit. I dont know you.
However, the only way you will be able to babysit is if your mom says its ok. So, like I said earlier, I think the one person you need to talk to about this is your mom. And, accept what she says without getting angry. I will be a good first step towards showing her you are mentally able to handle watching a child or children
Your mom probably thinks you're not ready for it, which is pretty understandable since you're only 13. Try to prove her wrong. Give her reasons why you want to babysit, and how you would be a good babysitter. Tell her how you would handle certain situations. Who would you be babysitting for? A neighbor? If so, tell her that you would be just across the street and that she could always check up on you to see how everything's going. If your mom still doesn't want you babysitting yet, then I would just respect that. She knows you best. I mean, babysitting isn't always fun and easy. Serious life threatening things can go wrong, and you have to know how to handle them. Trust your mom's instincts and try again next year.
Are there any young family members who you can babysit at your house? if you can bring a child home, your mother can actually watch how you deal with things, and since she's right there, there's no risk. If she sees that you handle everything well, she may be more ready to allow it.
Ask to have a serious discusion with ur parents about how they see things panning out in the comming years.
Tell her it's for training And that u r mature
By lettin her knw u're matured enough..
Show her how mature you are.