Odd one. It's definitely outside of the Ambury Park official boundaries, per Auckland Council GeoMaps.
It is Watercare land, and they don't really seem to name it. See their map of the walkway here: https://wslpwstoreprd.blob.core.windows.net/kentico-media-libraries-prod/watercarepublicweb/media/watercare-media-library/coastal-walkway/watercare_coastal_walkway_flyer_web_may2020.pdf