According to ITA these are the only flights from AKL to USA with 4 seats available for purchase in economy:


Air New Zealand 7:35 pm
Mar 27 3:10 pm 13h 35m AKL to IAH

Air New Zealand 7:35 pm
Mar 29 3:15 pm 13h 40m AKL to IAH

Air New Zealand 10:15 pm
Mar 28 7:50 am 8h 35m AKL to HNL


If the family can split their travel 1 adult and 1 child per flight then there are also these flights in premium economy:

Air New Zealand 7:45 pm
Mar 28 12:00 pm 12h 15m AKL to SFO

Air New Zealand 10:45 pm
Mar 29 2:55 pm 12h 10m AKL to LAX

AA's last flight AKL-LAX is on the 29th but the only availability this week is on the 29th with only 3 seats selling with 1 in Business and 2 in Premium Economy