Own Policy or Parents â€“ Adding your child to your existing policy will save them money on the amount that they have to pay. The downside of this is that if they get into an accident or get a ticket for a moving violation, your insurance will go up as well. If you do plan on adding your child to your policy, it is still a good idea to get a quote from a company for yourself and your child. That is one of the best parts about cheap car insurance for young drivers. You may find that you can save money on your car insurance as well.