The National Parks in SA are run by SANparks. At the end of this is a link to show you where they are.
By far and away the best known for 'safari' experience is Kruger which is on the Mozambique border in a low river plain - just the sort of thing mosquitoes love, so you need jabs there.
An alternative (that jno refers to) are the parks in the south of the country, of which the Addo Elephant Park north of Port Elizabeth is the one I'd suggest. You won't see all the Big 5 there - no lions or leopards. But the other 3 are there plus a host of other game animals including giraffes.
The second thing you need to understand is how the game parks work. Though they are National Parks a bit like our national parks - the Lake District say, whilst some of them are 'open' to anyone who wishes to drive through, vast areas of land are leased to commercial organisations who set up 'private game parks' where they have sole access to the land within those areas. So whilst Eccles talks about Kruger being a glorified safari park he is perhaps talking about the open areas where there is a free-for-all of private individuals and private independent commercial operators all trying to access the same animals. A scrum can occur.
But both Kruger and Addo have private game reserves - often on the 'best' land where the most animals congregate. The animals don't know any different and can wonder accross the 'boundaries' between leased and open-access area lands.
The private game reserve operators also sometimes lease additional land (and this certainly happens at Addo) where they can 'contain' (and I am talking areas areas of perhaps 50 square miles of fenced land) other game animals only found in Kruger - so I have seen white rhino, giraffe and many other species in such areas.
So you pays your money and takes your choice.
Kruger is perhaps far enough north to be month independent, but the parks in the south are best in October-December, I reckon.
http://www.sanparks.org/tourism/map/