Regulations don't necessarily encourage density, true (this is not always a bad thing though) - Victorian terraces with no parking, garden, space to store the bins etc wouldn't get planning permission now. On the parking - residents and the highways authority always complain if they think there's any risk people at the new development might park on the street.
But apart from that, there are other obstacles - obviously brownfield sites are more expensive to develop. In much of the country small houses and flats in a central location are not at all highly sought after, and those sites which remain undeveloped are clearly not the most attractive investment propositions. High rise city centre development is not usually the business of the volume housebuilders. On greenfield or other edge of town sites a development without parking a) (rightly) won't get planning permission in such an unsuitable location, and b) it wouldn't be worth so much, because you'd get all the disadvantages of lack of space without the access to services existing centrally located properties have.
I'm not suggesting that there should be no parking, rather that there doesn't need to be as much parking as there is.
If you've got four bed houses which are near (within 800m) of a primary school, local shops, nursery and a health centre. Not that far from (within 2km) of a secondary school and on a bus route (at least 2 buses an hour) as well as having good cycle facilities why should that house need to have 3 parking spaces when a house built out in the sticks with none of those facilities also has 3 parking spaces?
Likewise why should the local center, which has the primary school, local shops, nursery and a health centre provide all the required parking spaces for each of those as if they were provided independently of each other as well as all the visitor parking for the flats there when:
- parents with children at nursery are likely to also have children in the primary school
- most of the children will be within 800m of the school with half not needing to cross a main road and the other half who do having a traffic signal controlled crossing to help them
- peak demand for shops rarely coincidence with peak school parking and peak visitors to flats, with peak school parking being fairly short lived
- flats are most likely to have visitors of an evening when the nursery is likely to be closed and the staff have gone home
- again there is fairly good cycle and bus provision, meaning that a car is less essential
As an example when I lived near a school we often suggested to people who were visiting to park in the area of parking outside the school grounds when they were visiting us, as they were always doing so outside of school hours.
Likewise I used to work in an office which shared a car park with flats, the owners of the flats often used our spaces outside of office hours whilst our visitors used the shared visitor spaces during office hours. There was only 1 time this was a problem and that was when one of the residents was ill and hadn't moved their car. Even then it wasn't really a problem and there was plenty of other spaces which could have been used if needed without being a problem. That was in a 10 year period in that office.