Boil the tortellini: Bring a large pot of salted water to a boil. Cook tortellini according to package directions until just tender.
Do not overcook.
Rinse and cool: Drain and rinse tortellini under cool water to stop the cooking. Shake off excess water and spread on a sheet pan to cool completely.
Make the dressing: In a jar or bowl, whisk olive oil, red wine vinegar, Dijon, garlic, oregano, basil, red pepper flakes, and sugar or honey. Season with salt and pepper.
Taste and adjust acidity or sweetness as needed.
Prep the mix-ins: Halve the tomatoes, dice the cucumber, slice the onion, chop roasted red peppers, and slice olives and pepperoncini. Halve the mozzarella balls.
Toss the salad: In a large bowl, combine cooled tortellini with tomatoes, cucumber, onion, roasted red peppers, olives, pepperoncini, mozzarella, and parsley. Pour over about two-thirds of the dressing and toss gently to coat.
Finish and rest: Add grated Parmesan and more dressing as needed.
Let the salad rest for 15–20 minutes so the flavors meld. Taste and adjust salt, pepper, or vinegar before serving.
Serve: Enjoy chilled or at room temperature. Garnish with extra parsley or basil and a drizzle of olive oil if you like.