Yes - you do need permission to have a jumping castle or inflatable device on public land.
You will need to complete an Event Application Permit Form and Indemnity Form.
You will also need $20 million Public Liability Insurance that is separate to the Jumping Castle's Public Liability insurance along with a risk assessment that outlines the supervision of jumping castle, demonstrates that it meets Australian Standards, set up of the inflatable item and advice to wind management in accordance with the manufacturers guidelines.
You must understand your occupational health and safety duties when operating land-borne inflatable devices. For further information please visit WorkSafe.