Hi sonay,
So the link provided to the form seems fairly self explanatory but just in case there is any confusion.
My understanding in South Australia is that to change the child's name, given there are 2 parents on the birth certificate, that you would still need to seek consent from the father and submit a joint application.
Your son, at age 13, would also likely need to be interviewed to ascertain if he is mature enough to understand the consequences of a change of name (this step seems to be a South Australian thing?). If it is determined that he is mature enough (subjective and based on interviewers opinion), then he must also consent for the application to succeed. If it is determined that his maturity is not at a level to provide consent, you may still be able to change his name.
If the father does not consent to the change of name, and will not sign the joint application, then you would need to apply to the Court. If this is the path you need to travel then you would be seeking an order that the Court directs Births, Deaths and Marriages to change the child's name to 'such and such name'.
Good luck