The Verdict: Not eligible
Your partner cannot support a partner residence application until they are a New Zealand citizen or resident. Look at their own route to residence first, and whether you can be included in it.
A supporting partner on a work, student or visitor visa does not qualify, however genuine the relationship. This is about their status, not your relationship.
If your partner works in New Zealand, their route is usually the Skilled Migrant Category or the Green List; the skilled migration quiz covers both. You can often be included in their residence application as their partner, which is usually faster than applying separately afterwards.
Meanwhile, the partner of a work visa holder may be able to get a work visa of their own, which keeps you both in New Zealand.