I was surprised travelling across the US at a cultural difference - here in Australia, a 'national part' is a non-developed or little-developed area of land for everyone, free to enter, usually with campgrounds. In the US, a 'national park' is often a non-trivial payment to enter, with airconditioned visitors centers, good roads, staff everywhere, and sometimes even hotels and restaurants... Not all national parks are like that, but the first few I visited were. Bit of a shock.
I learned later in my trip about 'public lands' in the US, which were (basically) unmaintained and free for all to use, which was closer to national parks back home.
National Parks in the US have to meet four specific criteria to be granted the designation:
1) It is an outstanding example of a particular type of resource.
2) It possesses exceptional value or quality in illustrating or interpreting the natural or cultural themes of our Nation's heritage.
3) It offers superlative opportunities for recreation, for public use and enjoyment, or for scientific study.
4) It retains a high degree of integrity as a true, accurate, and relatively unspoiled example of the resource.
In the whole USA there are 59 such sites. But there are also national forests, national monuments, national seashores, national reserves, national historic sites, national historic landmarks, and other similar designations.
Our state forests system sound similar to what you mentioned from Australia. We had a couple of pretty huge ones near where I grew up:
You must be from the east. In WA, there are massive tracts of land that you can't get through, privately owned. This of course doesn't matter much - who wants to get to the Gibson, anyway, but .. its not like Australia doesn't have its own private owners.
I am not aware of any free-to-use national parks in NSW. There is a fee even for day use at the Royal National Park. I think State Parks are free, but I have had to pay whenever I have camped in NSW or Tasmanian National parks.
It is possible to buy an annual pass for all parks in NSW for a reasonable price.
I learned later in my trip about 'public lands' in the US, which were (basically) unmaintained and free for all to use, which was closer to national parks back home.