You can get a passport at any age with parental consent. Otherwise, you usually need to be 18 years old.
If you are under the age of 18, then you cannot apply for a passport without your parent(s) consent or the consent of all people with parental responsibility for you. If you are a child in care, then consent from the Department for Children & Family Services will be required under the Australian Passports Act 2005.
A passport cannot be issued against the wishes of the persons with parental responsibility over the child.
However, if you are living independently of your parents or person with parental responsibility and are not a child in care, then you may be able to show that ‘special circumstances’ apply that mean you may be granted a passport without consent of person(s) with parental responsibility.
This will involve filling out an extra form with the Passport Office. A Senior Officer will then investigate whether there is a good reason for not requiring your parent(s) consent (e.g. abuse, court order, etc.). There is no guarantee that your application for a passport will succeed. You will also need to provide proof that you are independent so will need to show you have an individual Medicare card, rental payments, pay slips and utilities in your name (electricity, telephone).
More Information
Australian Government, Department of Foreign Affairs & Trade, Australian Passports Office: