Preheat oven to 400 degrees F. Brush or spray a 1 1/2 quart baking dish with olive oil or cooking spray; set aside.
To make Alfredo sauce, melt butter in a medium saucepan over medium heat. Add garlic and cook until fragrant, 1-2 minutes. Add heavy cream, bring to a simmer and cook until slightly reduced and thickened, about 5 minutes. Stir in 1 cup parmesan cheese; set aside.
In a large bowl, combine shredded chicken, chopped broccoli, 2 tablespoons chopped parsley, 1/2 cup shredded mozzarella cheese, salt and pepper. Add 1 1/2 cups Alfredo sauce and fold to coat chicken and broccoli.
Spoon 1/4 cup Alfredo sauce into prepared baking dish and spread to coat. Fill shells with a large spoonful of the chicken and broccoli mixture and arrange in dish. Ladle remaining Alfredo sauce over shells, then top with 1 cup shredded mozzarella cheese and 1/2 cup grated parmesan cheese.
Bake uncovered for 20 minutes until sauce is bubbling. Garnish with 2 tablespoons chopped parsley and serve.