To be legally allowed to work in Australia, all overseas visitors and immigrants must hold an appropriate work visa, regardless of their qualifications. Check the following government website for further details:
https://immi.homeaffairs.gov.au/visas/working-in-australia
Once you have your visa, Pilates is not a regulated industry in Australia so you are not required by law to be registered or licensed with a professional organisation in order to instruct. However, many instructors find that joining the PAA gives them credibility and shows their commitment to the Pilates profession in Australia. It is also a useful source of support and advice for members.
You will also, in most cases, be expected to hold a comprehensive First Aid qualification, (i.e. Provide First Aid HTLAID003) and have Professional Indemnity and Personal Liability insurance cover.