In a bowl, mix melted butter, minced garlic, Italian seasoning, and salt. Stir in half of the chopped parsley and save the rest for later. This garlic butter will add strong flavor, so spread it evenly.
Cut the package of rolls in half horizontally without separating them. Make small slits between each roll, cutting about three-fourths of the way down. This keeps the rolls connected at the bottom for easy serving after baking.
Place the bottom half of the bread, cut side up, on a lined baking sheet. Brush about two-thirds of the garlic butter over the surface, making sure it gets into each slit for maximum flavor.
Sprinkle the mozzarella cheese evenly over the buttered bread, pressing it into the slits. Add grated Parmesan on top for added saltiness. If you like spice, sprinkle red pepper flakes now.
Put the top half back on and brush it with the remaining garlic butter. Cover loosely with foil and bake at 350°F for about 15 minutes. Then, remove the foil and bake for another 5 to 7 minutes until the cheese is melted and the top is light golden.
For a crispier top, switch the oven to broil for the last 1 to 2 minutes, watching closely so it doesn't burn. After baking, sprinkle the remaining chopped parsley on top and pull the bread apart while it's warm.