In the U.K., dental care is available on treated on the NHS just like other forms of care.
All non-emergency health in the UK is mediated via your GP. Except dentistry. All expert medical procedures in the UK will be carried out by an MD, no matter what part of your body they're on. Except for your teeth. For any serious medical treatment in the UK that requires anaesthesia, or surgery, you are likely to be treated at a general hospital which happens to have an appropriate unit. Except ... dentistry, where they have specialist dental hospitals.
Finally, treatment on the NHS for all medical conditions is free at point of use, except for prescriptions. Except dentistry, which is merely subsidised, and generally provided by dentists who work privately, and offer to provide slightly different treatments on the NHS than they do privately.
Point of pedantry, which doesn't really affect your argument: the professional medical degree in the UK is the MBChB (Bachelor of Medicine / Bachelor of Surgery -- abbreviation can vary by institution). The MD is either a research degree (similar to a PhD) or, at certain older universities, a higher doctorate similar to a DSc. In either case, most British doctors don't have one.
Dentists can take a mix of private and (public) NHS patients or specialise purely in private or NHS. If you move to a new area it can sometimes be hard to get into a practice as an NHS patient.
NHS dental care is heavily subsidised but not entirely free (See here for prices where I am in Scotland - https://www.scottishdental.org/public/treatment-charges/)
Typically, if I get a check up and need nothing done other than scale and polish the cost is around 15 USD (The exam part being free)
As a consequence, having some dental insurance plan is not uncommon here - my partner needs more dental work than I do so has such a plan called Denplan.
As one can imagine some grey areas can make it difficult to determine whether a dentist treatment is covered or not.
I guess parent is a bit more upper-middle class than you.
Edit: Pretty sure I pay £30 odd a month and I've had root canal treatment, new crowns, etc.
If you need a simple crown that is subsidized if not completely by the NHS you'll be able to offload it to the insurance pretty easily, if you want the top of the line ceramic crowns that cost 1500-2000 GBP no insurance will actually cover that.
For this I get 4 cleans/checkups a year and a zealous hygiene regime. I also get some insurance for accidental trauma too.
Sure, if you do need major work it's going to be expensive but the emphasis is very much aimed at preventing this. In the past 15 years or so I have had minimal work done.
Relying on an NHS practice is, as indicated in other replies, a bit of a lottery and you'll be hard pressed to get decent treatment for something serious. Partly because some dentists are not qualified to do stuff like root canal and partly because of funding restrictions - they just don't want that kind of work.
Having said that, things have improved since my childhood when NHS dentists were payed per filling. You can imagine how that worked out in many case.
Just like that. No new patients. Tough.
It's only relatively recently that you'll find yourself driving past one with "Now accepting NHS patients" on big banners outside.
I had a lot of NHS work done last year. I didn't pay much because I was used as a guinea pig by a couple of freshly qualified dentists who wanted to improve their skills before being promoted to private practice.
One dentist was good, one was kind of okay, I guess.
The deal wasn't explained. It was "And this is the NHS dentist you'll be seeing." Which is maybe not as ethical as it could be.
I've paid for private care in the past and it's never been significantly better. You get more choice of filling/crown material, but the actual work seems to be as good - or bad - as NHS work.
By far the best work was done for free by supervised students at an NHS teaching hospital in the 80s. Most of it has lasted for decades.
The issue with dentistry is possibly that there's no distinction between primary health care - the "have this tooth out or die of sepsis" kind - and purely cosmetic surgery, which is strictly optional.
There's also some overlap with products like implants, which improve looks but can also work better than fillings/crowns.
I can't think of any other branch of medicine which has the same tension between health and appearance.
In Japan, basic dentistry is covered the same under national health insurance, but for instance if you want a ceramic crown instead of metal, you have to pay for that yourself. Dentists and general medicine are completely separate institutions though.