Secure long-term parking in Johannesburg

It's worth noting that one reference on the Lonely Planet forums reckons the South Gate parking is under shade netting - which in Joburg's hailstorms may not offer the protection you desire.

However, your link would seem to indicate that's only sections P5 and P6.

Another option would be to rent a storage garage big enough to fit your car. They have 24 hour security, and there are some near the airport.

Finally, you could consider Omni Park - who manage several car park locations in the Cape (I realise that's a different area), and may have contacts or branches in or around Joburg.