Not sure how much of this you already know, but some information I found from searching using https://bdmhistoricalrecords.dia.govt.nz/Home/, which normally has a bit more detail than Ancestry does:
Robert's birth:
Registration Number 1918/1978
Family Name Taylor
Given Name(s) Robert Edward Conway
Mother's Given Name(s) Muriel
Father's Given Name(s) Conway Morten
Robert's father's birth (not sure when or why he changed his middle name from Whitfield to Morten):
Registration Number 1893/4385
Family Name Taylor
Given Name(s) Conway Whitfield
Mother's Given Name(s) Emily Charlotte
Father's Given Name(s) Herbert
Probably Conway's parents marriage:
Registration Number 1891/3631
Bride's Given Name(s) Emilie Charlotte
Bride's Family Name Murphy
Groom's Given Name(s) Herbert
Groom's Family Name Taylor
I can't find any record of Robert's father Conway marrying his mother Muriel in New Zealand. However since he enlisted in WW1 a slightly speculative possibility for the marriage might be:
Name Conway M Taylor
Event Type Marriage
Registration Quarter Apr-May-Jun
Registration Year 1916
Registration District Chertsey
County Surrey
Event Place Chertsey, Surrey, England
Spouse Name (available after 1911) Kirk
Volume 2A
Page 155
Line Number 133
Conway M Taylor probably married one of the following people
Name Muriel Kirk
"England and Wales Marriage Registration Index, 1837-2005," database, FamilySearch (https://familysearch.org/ark:/61903/1:1:26XJ-D9J : 22 May 2014), Conway M Taylor and null, 1916; from “England & Wales Marriages, 1837-2005,” database, findmypast (http://www.findmypast.com : 2012); citing 1916, quarter 2, vol. 2A, p. 155, Chertsey, Surrey, England, General Register Office, Southport, England.