The Verdict: Clear Path
You are already a New Zealand citizen, by birth. Apply to Internal Affairs for a New Zealand passport, or a citizenship certificate, as proof.
A birth in New Zealand, the Cook Islands, Niue or Tokelau before 1 January 2006 made you a citizen whatever your parents' status. All four are part of the Realm of New Zealand, so this is citizenship by birth, not by descent, and there is nothing to register.
Internal Affairs will want your full birth certificate and, if your name has changed, the documents linking your current name to it.
One narrow exception: a child of a foreign diplomat posted there at the time did not become a citizen. If that was your parents' position, say so when you apply.