Add green olives, red peppers, and giardiniera to a food processor, and pulse for 1-second intervals, about 6 times, until all pieces are about ¼ inch. Transfer to a medium bowl, and mix in minced parsley, capers, and garlic. Add vinegar and olive oil and stir until evenly combined into a spread. Cover, and refrigerate for 2 to 8 hours.
Use a serrated knife to cut the muffuletta roll in half as you would an English muffin. Give the olive spread a quick stir to incorporate all the vinegar and oil, and spread half on each piece of bread.
On the bottom half of the bread, layer half the sopressata, half the capicola, half the mortadella, and half the provolone. Repeat the layers, then top with the other spread-slathered piece of bread. Press down firmly, cut into quarters, and serve.