Heat butter in a large heavy skillet over medium heat; cook and stir onion, green onions, carrot, celery, and garlic until fragrant, about 5 minutes. Cover skillet with a lid and cook, stirring occasionally, until carrot is tender, about 5 more minutes. Stir 1/3 cup ketchup, Worcestershire sauce, salt, and pepper into vegetable mixture; cook until warmed through, about 1 minute.
Mix ground beef, sausage, vegetable mixture, bread crumbs, parsley, and eggs in a large bowl; form into a loaf and place in a rectangular baking pan with 2-inch high sides. Cover loaf with remaining 1/3 cup ketchup.
Bake in the preheated oven until no longer pink in the center, about 1 hour. An instant-read thermometer inserted into the center should read at least 160 degrees F (70 degrees C).
