Operational cemeteries in New South Wales are required to maintain burial registers. The information to be retained in the registers is defined in the Public Health Regulation administered by Health NSW Other jurisdictions will probably have similar regulations or laws.
Current requirements under the Public Health Regulation 2012 are here.
Ostensibly registers prescribed under acts of parliament should be open and transparent to the public. However, opinions and policies vary between burial authorities.
National Trust (NSW)
The National Trust has a list of cemeteries in New South Wales, excluding private family cemeteries and lone graves. The list is here.