adelaide rental no pets on lease

according to this

https://www.sa.gov.au/topics/housing/renting-and-letting/renting-privately/during-a-tenancy/pets-in-private-rentals

i can apply for a pet and the landlord can't reject my request?

i asked the landlord and they said there is not animals allowed on the lease. i'm not on the lease.

should i wait a couple weeks for a response, then file a request for a pet?