Hello...
As an alternative you could always try and contact a descendant of Joseph and Elizabeth and see if anyone in their extended family has already done the hard work.
You could try the following...
Found at BDM NZ Search
https://www.bdmhistoricalrecords.dia.govt.nz/search/BDM NZ
Births1908/5842 - GILMOUR, William Edward - mother:Elizabeth father:Joseph d.o.b
12 Feb 1908BDM NZ
Deaths1990/44233 - GILMOUR, William Edward d.o.b
12 Feb 1908 d.o.d 16 Jul 1990
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Found at Palmerston North Library "Manawatu Standard Newspaper Index"
http://ipac.citylibrary.pncc.govt.nz:8080/ipac20/ipac.jsp?menu=searchIn the box "Name Browse (for births, deaths, etc)" type Gilmour William Edward
Note down the details for the marriage and death notices of William Edward GILMOUR (subject, date of article, page number etc).
As the death notice was published on the 17 July 1990 it's safe to assume it relates to BDM NZ Death 1990/44233 - GILMOUR, William Edward
Go to the Palmerston North Library contacts page here ....
http://citylibrary.pncc.govt.nz/staff-contacts.htmland send an email to the "New Zealand and Pacific Islands Librarian" asking for a copy of the two notices, giving the necessary details.
She is really helpful and will email you the notices usually within a day or two.
Assuming there are any :-) the death notice should mention children of William (i.e grandchildren of Joseph and Elizabeth). You can then check electoral rolls or similar (google, white pages, old friends, fb) and make contact. Then just ask if they know of anyone in the family with an interest in family history and, if so, would they mind pasing on your contact details to them.
Sounds a bit convoluted but why re-invent the wheel when a few emails could do the trick.
Or if you want I can email the library, get the notices for you and then just forward them to you :-)
Regards
Beg