Will a parent's insurance go up with a new driver in the household if the new driver has their own insurance?

PROBABLY not if they specifically EXCLUDE that driver on their own policy. However, this will be an underwriting function and the underwriter could indeed raise the rate even if the parents exclude him/her. If the new driver has their own policy, though, the parents policy would AT MOST, only provide excess coverage. This excess coverage COULD (and probably WOULD) become available if a loss exceeds the policy limits of the new driver. So this fact may prevent the parents' rate being raised. HOWEVER, there is still the fact that, given there is another vehicle in the same household of the new driver, the likelihood that at some point in time they will drive that vehicle is really considered quite high.