確か、Bookinｇに使ったクレジットカードが必要だったと思います。今、ぐぐってみたら、Q and A のParking にこのように書いてありました。

https://www.aucklandairport.co.nz/frequently-asked-questions

What do I do at the airport on the day of travel?
Drive to the car park you have pre-booked. Signage is clearly visible at the airport, all car parks are identified by a letter. Please bring the map that is available on the website and on your confirmation email. Once you have located the car park you have booked, tap or insert your nominated credit into the slot at the entrance barrier. The credit card recognises you have pre-booked and paid for your parking and will open for you. Please remember that you will need to bring your nominated credit card to both enter and exit the car park.

予約確認のmail　と予約時に使ったカードが必要なようですね。Q and A　のところにいろいろ書いてあるので一度確認されてもいいと思います。