Only some children moving to New Zealand need exemptions to be home educated. Check out this link from the MoE:
My family has just arrived in the country and we are applying for residency. We would like to homeschool in New Zealand, but we are unsure if we need to apply for an exemption for our children.
In this situation you will first need to determine if your children have ‘domestic student status’. If they do and they are between the ages of 6 and 16 years of age, then they must attend a registered school unless they have an exemption.
If your children do not have domestic student status then they will qualify as foreign students, and you do not have to apply to homeschool them as foreign students in New Zealand are not required to enrol or attend a school.
To determine if your child or children are foreign or domestic students, please read the following:
What’s the difference between a foreign and domestic student?
Under the Education Act 1989 a foreign student is anyone who does not fall into the category of domestic student.
A domestic student is defined as:
At any time, means a person who is then –
a New Zealand citizen; or the holder of a residence class visa granted under the Immigration Act 2009 who satisfies the criteria (if any) prescribed by regulations made under subsection (4); or
a person of a class or description of persons required by the Minister, by notice in the Gazette, to be treated as if they are not foreign students.
These categories are explained in Ministry of Education Circular Eligibilty to enrol in New Zealand schools. The link to this circular is: www.minedu.govt.nz/circular201201
This information is found in Appendix E of the exemption form online: http://www.minedu.govt.nz/Parents/AllAges/EducationInNZ/Homeschooling.aspx