Before your child starts to learn to drive you need to have a conversation with them to agree on what restrictions will apply once they pass their test. Many parents contribute towards the cost of getting them on the road for the first time - driving lessons, purchasing a car and insurance to name a few. If you are going to contribute towards the cost you have a right to ask for something in return!
There's useful information about my road safety professionals are calling for a graduated driving licence here. But whether a GDL comes to fruition or not, there are steps each parent and new driver and their passengers can take to ensure their safety. Supporting new drivers in Great Britain
There are also great resources out there to create your own tailored parent/new driver agreement to cover all the main risks as well as others that are more specific to you and your family. RoSPA has a great free example here
RoSPA's Safer Driving Agreement between Parents and Teens