Preheat oven to 500 degrees. Stir together cheeses, garlic, and oil in a small bowl; season with salt and pepper.
Divide dough in half; let rest 10 minutes. Brush a baking sheet with oil. Stretch one piece of dough into a 10-inch round on sheet. Dot with half of cheese mixture, then top with half of pickles and half of soppressata.
Bake until crust is golden brown and cheese is bubbling, 10 to 12 minutes. Drizzle with oil, top with parsley, and serve. Repeat with remaining dough and toppings.