"Under the Family Law Act 1975, both parents — married or not — generally share equal parental responsibility for their child. This means both have the right to make decisions about their child’s welfare, education, healthcare, and upbringing.'
" In practice, the court’s main concern is the child’s best interests. Whether parents are married, in a de facto relationship, or separated, the same principles apply when determining custody or parenting arrangements.
If both parents are listed on the child’s birth certificate, their legal rights are recognised equally."
The above is from a New South Wales law firm. However the same applies throughout all states of Australia. See Relationships Australia website.