This came up in a facebook discussion I was having 3 months ago on drones.
Is it not part 101 of the Civil Aviation Rules that's relevant here? https://www.caa.govt.nz/assets/legacy/Advisory_Circulars/AC101-1.pdf#page=11
I don't have time to re-read it in detail right this moment, but from memory as well as dealing with land-owners being required to give consent, page 13 has a particular section on public land and spaces. It states that local authorites and DOC should be engaging with operators and providing necessary consent over the land they're responsible for.
Thanks for the link @izogi. For those in a hurry caa have a summary here:
And a faq here which includes an answer to the very question being asked:
If anyone thinks this situation will improve before it gets way worse; if anyone thinks the increasing numbers of people in the bush, especially international visitors, are going to adhere to their 'better angels' and not be increasingly clueless, inconsiderate, and obnoxious; if anyone thinks technology such as this will not compromise the wilderness experience for everyone: you're dreaming.