Looking at the baptisms for Deptford St Paul, which is in the Greenwich area, there is the following - note the birth date
1906 Jun 27, Catherine, d/o Augustus & Caroline HURLEY, 34 Catherine Grove, General dealer, born 1906 Jun 10
There was another child to this couple in 1904, Charles Edward and the father is given as William Augustus.
Looks like the birth certificate is the first thing on the list, to confirm parents.
William Augustus HURLEY married Caroline Ann CARVIN on 6 Oct 1902 at St James the Great, Bethnal Green.
Ian C