Drain the pepperoncini and press between paper towels until dry to the touch. Preheat the air fryer to 400°F.
Lay a wrapper in a diamond orientation. Sprinkle 1/2 teaspoon Parmesan across the lower third.
Layer on half a slice of provolone, half a slice of ham, 3 slices pepperoni, a heaping tablespoon of mozzarella, a scant tablespoon of pepperoncini, and a pinch of oregano. Keep the filling in a compact log.
Fold the bottom corner up over the filling and press to remove air. Fold in the sides, roll tightly, and seal the top corner with water or a flour-water paste.
Spray the basket. Arrange rolls seam-side down with 1/2 inch between them and spray the tops.
Air fry 5 minutes, flip, spray again, and cook 4 to 5 minutes more until deep golden.
Rest 2 minutes. Serve with Italian dressing.