This is actually pretty contested. If you go by the seven “traditional” continents (North America , South America, Europe, Asia, Africa, Australia, Antarctica), New Zealand wouldn’t be on any continent.
It is, however, part of Oceania, which is a large geographic region (very similar to a continent) including Australia, New Zealand, and about 10,000 other islands in the Pacific Ocean .
The coolest answer to this question, though, is that New Zealand is on its own continent of Zealandia. Some scientists argue that what we think of as the islands of New Zealand are just the peaks of the mountains of Zealandia, and the rest of the landmass is submerged. This theory isn’t widely recognized, but it is pretty neat.