No. Big springs is a few miles further up the road and has overnight camping
Back of Timp near Aspen Grove also has camping, up the opposite direction or "north fork" if you will. You can backpack up from the Southfork Parks and camp out without improvements and sometimes no fire.
You can camp at Nunn's Park which is heading towards Orem. Its ofc to the right at Bridal Veil Falls exit. Im not sure if the campground is open yet.
There is no overnight camping in South Fork park there is only picnicking facilities.... Realistically as far as camping you'll either have to make the choice of going to Orem or to Heber City
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