The short answer is yes, despite what you've probably heard from the internet, you are allowed to take your drone to New Zealand and you can even fly in the national parks if you follow the process.
DOC website for recreational drone permits
https://www.doc.govt.nz/get-involved/apply-for-permits/recreational-drone-use/
DOC Map
http://maps.doc.govt.nz/mapviewer/index.html?viewer=docmaps
