Street parking in any capital city is at a premium.
There are metered car spaces and a number of public parking stations in easy proximity to Local Government House. For more infomation visit: https://www.cityofadelaide.com.au/transport-parking/parking/parking-in-the-city/
Please see below links to public parking stations FYI:
East End Car Park
8 Union Street, Adelaide
Enter and exit at 277 Pirie Street, Adelaide
Enter and exit at Sudholz Place, Adelaide (off Flinders Street)
Public Transport Options
Getting around Adelaide by public transport is quick, efficient and environmentally sound. Travel into the city by train, bus or tram.
Why not try Park 'n 'Ride?
Park 'n' Ride car parks offer participants a place to park their car and transfer to an Adelaide Metro public transport service to complete their journey.
There are 72 Park 'n 'Ride locations offering more than 10,500 parks across Adelaide. For further information about Park 'n Ride locations, please see the following link:
If you prefer to park on the outskirts of the city, there is an Adelaide Connector -Free Bus Service that runs along Hutt Street. The Route Map and Timetable for the City Bus can be found at the following link: http://www.adelaidemetro.com.au/Timetables-Maps/Special-Services/Free-City-Services