Can a 13 year old steer a jet ski with their parent right there?

Of course ... with their permission. There are no licenses required for watercraft of that type.

I'm sorry to disagree but it depends on the state.
In NY you cant drive one if your 35 you need to have a license. the permission is important too. so you have to check your state laws. now i dont know if the parrent is on the ski with them if they can?