kind of pixie. If the teacher is carrying a pupil as a part of their job, then they need what used to be called full class A insurance, which is use for social, domestic, pleasure, travel to and from work and the extra bit which is use as part of work. If they are carrying the child as part of their work and they have an accident then the insurance won't cover the child and the insurance company may refuse to pay out at all.
If the teacher is a friend of the parent and is doing a neighbourly favour then they don't need the "car use as part of work" extra to their insurance, but would need to be able to prove that it was a neighbourly favour, ie not done within working hours, not organised by the school and so on.
I have dealt with a similar thing when a NHS employee managed by me started doing errands and favours for a patient. She was, of course completely entitled to do as she wished with her spare time but we had to make clear to her that anything that she did was as a private individual and she, and the patient needed to understand that neither were covered by any kind of NHS insurance or responsibility.