"From what I understand you can camp at Peralta Trailhead (for a fee) or you can camp at Carney Springs Trailhead for no fee."
While I didn't see a sign specifically prohibiting camping at Peralta Springs, there are no campsites. There is a rough campsite (at least, a level spot off the road with a fire ring and one small shade tree) between Carney Springs and Peralta, on the N side of the road. The validity of this site is questionable; the property to the immediate N is marked "No Trespassing".
The Carney Springs Road is now closed to vehicular traffic. You can't camp at the springs unless you are willing to backpack.
"If you go all the way to the Peralta Trail there is a parking fee which you can pay with your credit card. "
I didn't see an obvious fee, and certainly didn't see any capacity for paying by credit card...