Put the olive oil in a small saucepan with the oregano, garlic and a pinch of salt. Warm over medium heat just until the garlic turns golden, then turn off the heat and let steep while you prepare the dough.
Preheat an outdoor grill over medium heat.
Unroll the pizza dough and roll it out by about 1 inch in either direction. Slice into 6 equal pieces. Stretch and press each piece into an approximately 11-by-2-inch rectangle. Sprinkle 2 tablespoons of mozzarella down the center of each rectangle, then fold the dough over the cheese and pinch the seams to seal, making 6 cheese-filled dough "tubes."
Thread each tube onto a skewer like a snake, alternating after every turn with a slice of pepper and a folded slice of pepperoni. Lay the pizza sticks on a parchment-lined baking sheet and brush all over with the garlic-oregano oil.
Grill the pizza skewers uncovered, turning every couple of minutes so that all sides make contact with the grill, until the dough is fully cooked, 6 to 8 minutes. Transfer to a serving platter, sprinkle with the Parmesan and serve with warm marinara sauce on the side.