Do the parents insurance go up if the child is 16 and gets a drivers license but does not drive the parents car?

If a child has a driver's license, the child has to be listed as a driver somewhere on an insurance policy. The child can have their own policy and then the parents rates would not go up. Usually it is less expensive for a child to be listed on a parents policy rather than getting their own policy. If the child truly is not driving a household vehicle than the only way for that child to not be rated is to turn in the driver's license. That should be fine since the child "isn't driving anyway". The child can still get a state I.D. that isn't a driver's license.

If the child isn't going to drive there is no reason to list them. The previous is correct, just should have gotten an I.D. and not a drivers license.

It may depend on the country or state. Where I am the insurance company would have no way of knowning if someone in you house got a drivers license without you letting them know.