Are Louisiana residents holding auto insurance policies covered on the
auto  policy if a minor dependant (age 15 to 18) living in the same
household and driving with permission of the policy holder is involved
in an accident if the minor IS NOT listed on the policy as principal
driver?

Most insurance companies require you to have anyone driving a vehicle
that they cover to be on the policy.  So in response to your question
No the driver would not be covered by your insurance.