I'm a bit confused about why Centrelink would determine a person to be a parent if that person is not listed as the parent on the child's birth certificate, but I'll provide some information nonetheless.
Nobody has any rights under family law except for the child. There are no parenting rights, per se, there is only a child's rights to know, spend time and communicate with parents and other persons significant to their care, welfare and development, on a regular basis insofar as their best interests can be met.
To pursue parenting orders, the man would have to prove to the court that he is a person concerned with the child's care, welfare and development and that parenting orders in his favour are in the child's bes interests. It would be a hard case to prove, given the brevity of the relationship. The child's age and nature of relationship with the man would also be influential.