Yes, that's the family.
Birth 14318/1886 GILLETT KENETH R - VINCENT - ELIZABETH - GOULBURN ** just 1886 in index ? ** No CofE baptism in Goulburn
Kenneth understated his age for some reason when he enlisted or there is an error in the BMD. I read it as 23 years and 1 mo.
Enlisted Liverpool, NSW 13 Sep 1915 - S/N 4298 2nd Bn > 54th >56th Bn - 23 yrs 1 mo ?  Labourer - NOK mother Gillett Elizabeth living at 'Woore' Goulburn, NSW - Embarked Sydney HMAT A60 'Aeneas' 20 Dec 1915 - Served in France - Returned to Australia per 'Wyreema' 3 Jun 1919 - Discharged 26 Jul 1919
It gets a bit tricky sorting out 3x Vincent.
GILLETT Vincent 23 'Ramillies' 19 Nov 1855 Moreton Bay Reels 2137, 2471
Vincent 23 - Somersetshire Samuel & Henrietta in Somersetshire CE Neither Sister in Sydney
Death 2954/1917 GILLETT VINCENT - VINCENT [?] ANN [?] - WAVERLEY ** 16 Mar 1917
GILLETT Vincent Death notice 16MAR1917 Death 86 late of Bondi SMH 17MAR1917
Parents should be Samuel & Harriet.