I suppose she can apply to the court for parenting orders that provide for her to have parental responsibility, but a court won't make an order to that effect unless it's deemed to be in the child's best interests. What a court considers where working out what's in a child's best interests is outlined in section 60CC of the Family Law Act 1975.
One of the components is the extent to which a parent has taken, or failed to take, the opportunity to spend time and communicate with the child, and another is the extent to which a parent has taken, or failed to take, the opportunity to be involved in the long-term decisions regarding the child's care, welfare and development.
Would a court deem it in a child's best interests to give responsibility to a parent who has thus far wanted nothing to do with them? Would that provide a stable environment for the child, or an emotionally secure one?
I would wager the mother would be granted time spent with the child, but probably not parental responsibility. Of course, that depends heavily on the evidence supplied for either party, as well.