Its both the parents and unfortunately the state of the UK. Teachers should not be responsible for overseeing toothbrushing!
I too am shocked at how many parents simply do not do or teach their children the basics..... brushing teeth, toilet training, weaning, washing, Reading/talking, playing with their children. Even taking them to a park or a walk is too much for many and then they wonder why their children are so hyped and 'bad'!
I think a lot of parents just can't seem to consider that their baby is a person that grows up, has their own needs and wants, and need to be taught how to do everything. They simply expect their children to somehow wake up one day and use the toilet.
But........, there are so many economic and political factors against parents. I remember seeing lot of adverts and being taught periodically in primary school about brushing twice daily for 2 mins. My parents took me to the dentist regularly fir check-ups. I dont think that's happening as much now. As PP mentioned, even if they do brush, children are constantly eating hidden sugars in foods people think are healthy as well.
People arent getting education from dentists as there's hardly any NHS dentists and the remaining ones have huge waiting lists because they simply cannot take more patients, including children. The £40mil spent on extracting teeth could be used to fund NHS dentists properly.
I can't remember the last time I saw an advert promoting brushing teeth. Electric toothbrushes, yes, but nothing specifically about brushing for 2 mins.
Public health promotion in schools/health visiting: They only give free toothbrushes and toothpaste if the company (eg colgate) send it to them for free, it is area based and for some reason the companies pick and choose what areas they want to give free samples to.
Basically, children's health in the UK is nearly the worst in the Western world. A lot of that is down to poverty, services being cut and chronically underfunded, the ease of UPFs in ready made foods, sweets, oven foods, etc. Whilst it's easy to blame parenting at an individual level, the bigger picture needs to be looked at because something is going terribly wrong.