Adopted persons have explicit rights to information about their birth family under the Adult Adoption Information Act 1985. These are set out in this website
https://www.govt.nz/browse/family-and-whanau/adoption-and-fostering/finding-your-birth-parents/
I recommend that this is the first port of call for all adopted persons, apply for their pre adoption birth certificate (sometimes referred to as an Original Birth Certificate or OBC). The form for this should be downloaded from the Department of Internal Affairs (BDM) website
An adopted person can also apply to
Oranga Tamariki for all identifying information held about a birth parent named in their pre adoption birth certificate
A Judge, to open the Court adoption records, if they have "special grounds,." Generally this information is about the adopting family, sometimes there is additional information about the birth family.
Additional information may be available from the hospital where you were born, or the solicitor who dealt with your adoption. This is usually obtained directly from the agency or solicitor involved.
An adoption person may also place a veto on his/her birth records.