HI Natalie,
As Sparrett suggested, you can search the Archives site using any search term you like. Even his date of birth, or Parramatta or other family members names to find the next of kin that he gave. You should be able to fill in the whole weekend looking for anything likely.
You said that his family sent him away to be raised by someone else, does your Dad have any idea where of by whom? The address of his family at the time, if you knew it, could be used as a seach term.
I think its more useful for you to be looking at these random records as you may recognise a name or place that the rest of us wouldn't.
mum mum