I've heard some countries let children travel as long they are with their parents who have valid passports. Is this the case for New Zealand, assuming the parents and child are Australian?
They can't go to New Zealand, or anywhere else for that matter, because children can't leave Australia without their own passports. From smarttraveller.gov.au:
All Australian citizens must have a valid passport before leaving Australia and maintain a valid passport while overseas. Under Australian law, children cannot be included in an adult's passport. All children travelling overseas, including newborn infants, must have their own passport.
Thanks to the spread of biometric passports, which are strictly for one holder, family passports that include children are pretty much obsolete these days.
Quoted from "People travelling to New Zealand - Information for airlines" from Immigration New Zealand:
A child travelling to New Zealand must:
- have their own valid passport, certificate of identity, or other travel document, or
- be included in the passport of the parent who is travelling with them, and
- hold a visa, if required.
So to answer the first part of your question: Children can (still) enter New Zealand without their own passport if they are included in a parent's passport and also travel together with that parent. If they are not included in the parent's passport, they need their own passport.
In the case of Australian citizens, jpatokal already explained why this is irrelevant, as Australia require children to have their own passports.