Is insurance required if you are only driving in a parking lot?

If it is a public lot then you are probably required to have insurance.

If it is a private lot then you may not be required. It all depends on the specifics of the situation which you have not provided.

Another point to consider...

A "private" lot doesn't usually mean what people think it means.

Check your state laws but in New York State, any lot that has a certain amount of traffic on it (like 10 cars) is considered public, regardless of who owns the land.