Australia will only require registration and probably pilot test for 250g and over.
Our system doesn't start for recreational flyers until sometime into 2020.
Until that happens, anyone can buy and fly a drone, sticking to CASA rules to fly, and various state agencies like National Parks, and local council bylaws in many metro regions.
Not sure how long countries with (bringing in) the same 250g rule will take to just drop the limit to 200g or whatever.
If DJI and / or other manufacturers keep developing sub 250g drones, they will probably want to get them under their watchful eyes.
As far as National Parks goes, yes NZ authority (DOC - Department of Conservation) is pretty hot on drone activity.
Understandable in many places.
We spend an entire February a couple of years ago walking tracks around Queenstown region, and the 'No Drones' signs are pretty well in all the visitor hotspots.
I think DOC is like some our various named Australian National Parks authorities, although here we have the 7 state / territory agencies, some are totally ok to fly under CASA rules, others have a fairly simple permit system, and a few are just totally no drones, even making commercial licenced pilots jump through many hoops to get a permit, along with a hefty fee.
Enjoy both countries flying your drone(s), there are many places outside controlled airspace and other banned areas to fly and film