There are some suggested answers on the Internet, but they are far from being accurate about who invented school in particular. Most of them point to the Byzantines. They were said to be the first people on earth to come up with the idea of a school system. These people continued the Roman Empire in the Middle Ages. They were heavily influenced by the Greeks, especially in language.
In 1369, a Mr. Harry P. School was said to start the idea of gathering naughty kids in the neighborhood and locking them up in a building. The idea seemed good to the parents. Later they employed an adult to look after the kids. And some say that started it. Mr. School was who started the idea of a school, the very man who invented school.