It depends what state the children live in. All states in the United States have "compulsory education" laws (which means all kids have to receive a formal education). The ages (or grades) kids have to be in school varies from state to state. Some states require that kids be in school from the age of five - others are as late as 8. Kids have to stay in school until they're 16-18 years old, again depending on what state they live in.
Parents who wish to home school their children have to work closely with the state to ensure that the students are being adequately supervised and are receiving an education that meets the state's standards.
Parents can be punished for failing to follow compulsory education laws. Punishments range from fines to community service to "hard labor" sentences.