Chop ingredients: dice salami, ham, provolone, peppers, onion, and olives small and combine in a large bowl.
Make dressing: whisk olive oil, red wine vinegar, oregano, garlic powder, salt, and pepper in a small bowl.
Dress mixture: pour dressing over chopped ingredients. Toss to coat. Add parsley if using.
Drain: spoon mixture into a fine mesh strainer. Let drain 5 minutes to remove excess liquid.
Prep rolls: split hoagie rolls open, press bottom gently to make a pocket.
Fill rolls: press chopped mixture firmly into each roll. Sprinkle Parmesan if using. Wrap tightly 10 minutes.
Serve: cut each sandwich in half and serve.