Place the roll bottoms on a baking sheet. Divide chicken, Cheddar cheese, barbeque sauce, onion, and parsley equally over the rolls. Cover with roll tops; brush with melted butter. Sprinkle Parmesan cheese and garlic on top.
Bake in the preheated oven until golden brown and cheese is melted, about 20 minutes. Slice and serve.
Hawaiian sweet rolls can be substituted for the dinner rolls.