In the Australian Constitution education is not listed as a federal government responsibility, which means it becomes a responsibility for state and territory governments. This means state and territory governments – particularly education ministers and departments – run schools and set education policy.
Here's where it gets a bit more complex. The federal Parliament is able to collect more taxes than states, so the federal government has more money than the states to spend on things like education. Although the state and territory governments run their own schools, they may need to follow federal government policies to receive federal government funding. Section 96 of the Australian Constitution allows the federal government to give money – 'tied grants' – to state governments based on certain terms and conditions.
This arrangement lets the federal government carry out important national education policies, like the Australian Curriculum. The education minister and bodies such the Commonwealth Department of Education and Training are responsible for this.
The PEO timeline has some more information about the history of school funding in Australia – see States Grants (Science Laboratories and Technical Training) Act 1964.