I don’t know what NSW tenancy laws have to say about pet restrictions that landlords might want to impose.
This from the TenantsNSW fact sheet on pets:
There is no term in the Residential Tenancies Act 2010 that prohibits you from keeping a pet, or that requires you to ask for your landlord’s consent before you keep a pet. However, many landlords will include a clause restricting pets in the residential tenancy agreement (i.e. your lease), and there is no specific ban on them doing so. The Tenants’ Union believes that such a restriction is a breach of your reasonable peace, comfort and privacy, however this has not been fully tested before a court or the Tribunal.
The standard form of the residential tenancy agreement issued by NSW Fair Trading includes additional terms which require you to have your landlord’s consent to keep animals. Additional terms may be crossed out when you and the landlord sign the agreement, but if they are not crossed out, they will apply to your agreement.
By the way, the Tenants’ Union websites in the various states are just about the only authoritative source of information for both tenants and landlords. Just click on tenants.org.au and it will take you to the relevant site for your state.