In a large mixing bowl, combine the shredded chicken, mayonnaise, Greek yogurt, Dijon mustard, red wine vinegar, garlic powder, onion powder, Italian seasoning, salt, and black pepper. Mix until well combined.
Fold in the diced celery, diced red bell pepper, sliced black olives, grated Parmesan cheese, and chopped parsley until evenly distributed.
Cover the bowl with plastic wrap and refrigerate for at least 30 minutes to allow the flavors to meld.
When ready to serve, slice the hoagie rolls in half and scoop a generous portion of the chicken salad into each roll.
Serve immediately and enjoy!