The New Zealand government invested ~ £750 million into fibre to the premises infrastructure, in partnership with other companies. About 80% of NZ homes have access, with say half of those actually using it, but percentage is growing. Usable plans are about £40 per month. In my city you can’t sign up to a cable connection, even if the house is already wired up for it. I belive you even have trouble signing up for copper, because the infrastructure companies just don’t want to support it. The main competitor left is mobile, although I would love to see StarLink be available here.
There are a lot more people in the UK though, housing is much denser and people are very resistant to change; it's easier to rip up the ground in NZ bc our country has only existed for 150 years, in the UK it's a lot harder with all sorts of caveats/consents.
Same as covid: Jacinda did an excellent job, but at the same time as a realistic kiwi, it's very true that NZ has a very small, spread out population and the benefit of being small and extremely southerly, international travel just isn't as prevalent there as say, Europe.
However actual usage is quite different. If your Internet seems fine, why would you buy a more expensive service that claims to be "faster"? In my city Toob are trying to aggressively acquire customers for FTTP with a price they're presumably losing money on, but I don't expect that to last.
The technology for a setup like your parents is either VDSL or G.fast. Near to their old (perhaps slightly battered) green BT cabinet is a newer one, the newer one has fibre to it, and a DSLAM, the DSLAM ties to the old cabinet with copper, and so only that "maybe half a mile away" distance is covered by the DSL technology, from there it's fibre.
In the UK, BTs 'fibre' rollout is almost exclusively FTTC and then copper phone lines after that. My parents have a phone line from 35 years ago, and the cabinet is maybe half a mile away, but they can get 50 Mbit down (I can't remember upload, maybe around 15 Mbit). The provider rolled this out a lot quicker than ADSL (we never got ADSL2), so I assume it's technically not that complicated.