Preheat the oven to 375 degrees F (190 degrees C).
Heat a large skillet over medium-high heat. Crumble sausage into skillet and cook until browned, about 5 minutes. Add onion, mushrooms, and celery; cook for 5 more minutes. Mix in garlic, sage, and rosemary; cook for 2 minutes. Turn heat off.
Whisk together chicken broth, potato flakes, and eggs in a large bowl until well combined. Add sausage mixture, Tater Tots®, and parsley. Mix well and pour into a 9x13-inch glass casserole dish.
Bake in the preheated oven until stuffing is cooked through and lightly browned, about 45 minutes. Turn broiler on high and broil until tots are crispy, about 3 minutes.