This would in effect be a parallel government for certain purposes.
Not under the Constitution. Section 56 provides:
56. A vote, resolution, or proposed law for the appropriation of revenue or moneys shall not be passed unless the purpose of the appropriation has in the same session been recommended by message of the Governor-General to the House in which the proposal originated.
Once again our Founding Fathers have demonstrated how carefully they constructed one of the world's most successful systems of governance.
Professor Anne Twomey answered this and other questions posed by Professor David Flint in a Conversazione held at Parliament House Sydney on 16 September, 2010