Preheat oven to 350°. Spray a 3-quart baking dish with cooking spray.
In a large braiser, cook sausage over medium heat, breaking up meat with a spoon, until browned, 7 to 10 minutes. Add onion and garlic; cook until vegetables are tender, about 10 minutes. Add broth, collards, and thyme; cook, stirring occasionally, until collards are tender, 10 to 15 minutes.
In a large bowl, whisk together cream, eggs, parsley, salt, and pepper until combined. Gently combine sausage-and-greens mixture, cornbread, and white bread. Transfer bread mixture into prepared baking dish.
Bake, covering with foil to prevent excess browning, until golden brown and set, 30 to 35 minutes.