I'm starting to wonder if there were two William Stanley's fathering babies in Deptford in 1704. 
Confusingly, the mother's name isn't mentioned on the register.
He must have become a Distiller between June 1704, when Henry Stanley was buried, and the 21st of September 1704, when Thurston Stanley was baptised in Deptford.
Dunne and Henry were baptised on the 14th of April 1704. So if the twins were baptised then, they must have been a good few months old at the time for the next one Thurston to be baptised just five months later.
According to his city of London admission paper from 1738, Thurston's father was indeed a Distiller, "late of Rotherhithe(?)", from Ancestry.co.uk.
So William must have obviously died before that date. There is a burial in Deptford from 1736 on Ancestry, of a:
"William Stanley, a Caulker."
A William Stanley married a Mary Webster at St Nicholas, Deptford on the 28th of July 1687.
Before the above mentioned Thurston, these children were baptised as children of Wm. Stanley at that church:
Charles - 24.04.1692
Mary - 03.06.1694 (possibly buried before 1706, because they had another Mary baptised at the same church in January 1706).
Elisha - Named "son of" on the original image on Ancestry. Transcribed as "daughter of", on their index. Bap. 11.05.1696.
Isabella - Bap. 21.07.1698.
Bur. 24.07.1698
George - Bap. 6.02.1699
Bur. 08.04.1700
