That 1913 death that Merlin has spotted may well be the elusive William's. The lack of parents names on the index does not actually indicate any other lack of info, as afterall, IF this is your chap, then he would be shown as a widower, and hopefully one of his children would be the informant, or someone who knew the names of late wife and their children, as there are provisions on Vic death certs for that depth of detailed information. That would be one way to help confirm or eliminate that William from your quest.
(PS, I think Vic BDMs are among the best certificates available from a family history viewpoint)
Cheers, JM