My first post addressed the documentation to show Customs when you arrive in the US. You will have to check with a NZ dealer to find out their rules on how to get the gun to the airport.
In the UK, my dealer escorted "his" guns to the airport. I do not have a UK shotgun certificate, so I could not take ownership in the UK or transport the guns to the airport.
My dealer transferred the guns to the airport agent to be screened and placed on my flight. I actually never touched the guns enroute until I was back in the US and getting them off the luggage carousel.
If it is just one antique gun and you have good documentation, I cannot see why shipping it to yourself or maybe your parents with a good tracking number isn't the easiest way.
Joe
Joe