If the Federal rules pertaining to working are followed, 13-year old kids can work. Kids at the age of 13 can do farm work on the family farm for example.
Thirteen is a young age to start working; however, there are many young kids who want to start making their own money. According to the Fair Labor Standards Act, a 13-year old child can only work on an agricultural job and must have signed permission or if the parent works on the same farm. The child can also only work during non-school hours.
On the other hand, there are many part-time jobs that 13-year old kids can do to make money that do not require government intervention. Many kids babysit for family friends and relatives, cut grass, wash cars, and other odd jobs that people in the neighborhood need help with. All it takes is a little creative thinking.