Thanks for the replies. It crosses my mind that there are two ways to read this sentence:
Economy:
Eligible for Airpoints Dollars & Status Points accrual (may not apply to sectors not operated by Air New Zealand).
No accrual of frequent flyers miles/points with airline partners.
It could mean, that you get no miles/points for flights on partner metal.
It could also mean that this sale is more restrictive and you get no miles/points for any of the flights in Economy (even on NZ metal).
I'm thinking that perhaps it does mean that if part of your flight is on a partner metal (ie United) that it does not earn any miles/points in the NZ program.
Still a little nervous about it, but thinking that perhaps the sentence is actually referring to the possibility that some of the flight could be on non-NZ metal and therefore not miles/points earning.