You really need to post a copy of the relevant bylaw to be sure, but my opinion is they can deny you permission.
Claiming the dog is a visitor is possible but it has been ruled (in QLD) that a visitor staying overnight is considered a temp resident and so not allowed to use the visitor car parking.
The reasoning behind this was short term letting in tower blocks to holidaymakers who were using visitor parking bays claiming they were visitors (which they were) resulting in no Visitor bay being available to genuine visitors. This same argument could be used throughout Oz.
If you are going to claim the dog is just a visitor you need to find out what NSW define as a visitor.