There is no Census information available for Australia - everything from 1901 was destroyed after the Census was taken for privacy reasons.
Voting in Australia is compulsory, so the electoral roll is a valuable place to look. He doesn't exactly have a unique name but you might get lucky and find him living near his cousins.
Unfortunately, the only way to access the electoral roll online for every available year is at ancestry.com
Of course, with all the libraries closed many are allowing you to access Ancestry Library edition from home.
The Queensland registry indexes include parent's details on death records, and if you buy a certificate you will receive it immediately as a pdf. See https://www.familyhistory.bdm.qld.gov.au/
Victoria is the same, see https://www.bdm.vic.gov.au/research-and-family-history/search-your-family-history
NSW does not provide as much detail, and you cannot receive the certificates immediately, except on rare occasions. See https://familyhistory.bdm.nsw.gov.au/lifelink/familyhistory/search?1
The Western Australian Indexes provide parents only for people who died after 1945 or so, and the index only goes up to 1971. To get a certificate you have to mail your request in, pay a fortune and have it mailed back to you. https://www.bdm.justice.wa.gov.au/_apps/pioneersindex/default.aspx
Trove is awesome, particularly with the digitised newspapers, and also links to various state records. See https://trove.nla.gov.au/
To be thorough though, you'd need to look up each state archive, especially for will and probate indexes. You'll usually need to pay for these, although Victoria has all wills prior to 1924 or so scanned and available to view for free. Not useful in this instance of course.
For burial records try austcemindex.com and billiongraves in addition to findagrave. For Western Australia ozburials.com has excellent coverage but the search function leaves a lot to be desired.