Not right but I think they've got it covered in the Act & Regulations?
SA Forestry Act 1950
21Regulations
The Governor may make such regulations as are contemplated by, or necessary or expedient for the purposes of, this Act. Without limiting the generality of subsection (1), the regulations may (a) provide for the control, management and protection of forest reserves and all other property belonging to, or under the control or management of, the Corporation; (b) prohibit or regulate access to, use of or conduct on land belonging to, or under the control or management of, the Corporation;
SA Forestry Regulations 2013
Part 2Entrance to reserves or parts of reserves
6Prohibition of entrance by notice
A person must not, without lawful authority, enter or remain in a forest reserve or part of a forest reserve contrary to the terms of notices erected with the approval of the Corporation at the entrances to that reserve or that part of the reserve. Maximum penalty: $750. Expiation fee: $105.
24Compliance with notices and signs
A person must not, without lawful authority, contravene or fail to comply with a direction by the Corporation regulating or prohibiting (a) the driving or parking of a vehicle in a forest reserve or part of a forest reserve;
Are there any signs up?
Don't know why they can't use permit system out of the regulations though? Maybe issue keys & permit with drivers license & a key bond/deposit of $50 or something.
26Display of certain permits in vehicles
(1) If a permit authorises the driver or passengers of a motor vehicle (other than a motor bike) (a) to enter or remain in a forest reserve; or
(b) to drive the vehicle in a forest reserve; or
(c) to camp overnight in a forest reserve,
a person to whom the permit applies must, when acting pursuant to the permit in a forest reserve, ensure that the permit is displayed in the vehicle in accordance with this regulation while the vehicle is in the reserve. Maximum penalty: $750. Expiation fee: $105.