Virgin Australia VA is not part of Star Alliance, but is a ff partner of some Star Alliance airlines & other airlines. You will need to check the Star RTW rules, but expect you cannot use these codeshares.
Normally such domestic code shares can only be purchased as a tag on flight to a flight operated by the airline (example PER-SYD-LAX on DL)
Air NZ fly PER-AKL & AKL-SYD, so via New Zealand will be the only way on a Star Alliance operated & marketed flight.
Or just buy PER-SYD with cash