Yes, every state (and the federal government) recognises de facto couples. However, every state does so differently, and every state has their own legislation about it. Some states have de facto relationship registers and others do not (NT and WA do not). Marriage law is federal; de facto law is mostly state laws, other than family law stuff which is federal. eg this article outlines how things are different in WA, with WA de facto couples not eligible to each others super (but you are in NSW)
What are the differences for de facto couples and married couples in Western Australia? (Also, the differences for de facto couples in WA and the rest of Australia!) - Family and Matrimonial - Australia
When you say de factos and parental rights, do you mean in regards to stepchildren? Because of course biological parents have rights in regards to their children no matter what kind of relationship they had or did not have. If you mean if de factos break up do the de factos have rights to their ex-stepchildren? According to this
https://emfl.com.au/what-are-your-custody-rights-in-australia-as-a-step-parent/ - married and de facto stepparents are equal in status, and how much parental rights they can be given depends entirely on the individual situation at hand, down to even sometimes having to pay child support.
