If you are the biological father, then you both have equal parental responsibility. This is the case unless a court order specifically removes or modifies it. Parental responsibility is not about who a child lives with or spends time with. If you can't agree on these things and enter into a parenting plan or consent orders, then a court can decide. Extremely unlikely that any one parent is given full time care with no time spent with the other parent. The family courts default position is that a child should spend enough time with both parents to form and maintain a meaningful relationship. This is excepted as being in the best interests of the child.
She will not be able to apply for orders without first participating in mediation to seek a plan on living and visitation arrangements. Perhaps you should call relationships Australia if she doesn't and start the process. If no agreement can be reached via mediation, then you will be issued with a Section 60I certificate that will allow you to apply for court orders
