Chris, if you use "Search" (this website - see top of screen) you will find past messages listing New Zealand resources, especially passenger records.

The most useful to you are probably the free b/m/d indexes through which you can order photocopies of certs (cheaper for a scan rather than a certified copy, when supplying the regt no.). These are far more informative than UK ones:
www.bdmhistoricalrecords.dia.govt.nz/Home/
and:
paperspast.natlib.govt.nz/cgi-bin/paperspast
which are full text newspapers online. These recorded passenger arrivals, as well as local news of deaths and obits, etc.

On the above website there is a birth for a William Keig in 1864: registration number 1864/3145; and an Ellen Jane Keig in 1868: registration number: 1868/4497

When you have their parents' names from these certs you can search for the parents' deaths. (The births also often have the exact place of birth of the parent who registered the birth.)

Sue