Not too many but you have a few options. You can babysit, walk dogs, tutor, offer to do yard work for your neighbors, or maybe try selling some things online that you don't use anymore. If you go with that last one, have a parent/guardian there with you. Obviously you can't get a real job but if you have a friend who's family own's their own small business maybe ask if you could just do some small things under the table for them.

do errands for the neighbors, like walk their dogs, wash their cars, rake leaves, or get their groceries. Ask each neighbor to refer you to another that could use your help (called marketing yourself). Also, sell what you have but no longer need or want on eBay.

Tell friends and relatives you want cash for birthdays and holidays.

work

you can open up in your parents name

begging and borrowing

I know that allot of the trouble with people moving out is that they don't save their money so that they can pay all the expenses. I want to start saving early, but what kind of jobs can a 13 year old get?