aussie airlines jump through zero hoops to operate in new zealand and vice versa. However, for an aussie airline to operate internationally from new zealand to beyond points, there are a few hoops they need to go through such as QF's AKL-LAX and NZ's SYD-LAX of which both routes cease to operate. As per the current bilaterals:
ANNEX
Section 1
ROUTE SCHEDULE
A. Routes for the airline or airlines designated by the Government of New Zealand
1. From points behind New Zealand via New Zealand, and any intermediate point(s), to any point(s) in Australia, and beyond.
2. For all-cargo services, between Australia and any point or points.
B. Routes for the airline or airlines designated by the Government of Australia
1. From points behind Australia via Australia, and any intermediate point(s), to any point(s) in New Zealand, and beyond.
2. For all-cargo services, between New Zealand and any point or points.
C. Routes for SAM Airlines
1. Between points in Australia and/or New Zealand.