Place the beef, lamb and chicken in a tall pot. Add the stock and bring to a boil. Reduce the heat until the liquid is very gently simmering (small bubbles should just break on the surface). Simmer, uncovered, 90 minutes, or until the meats are tender.
Using a slotted spoon, remove and set the pieces of beef, lamb and chicken on a baking sheet and cool to temperature. Cut meat into cubes and refrigerate until needed. Skim fat from the surface of the stock left in the pot (see Note).
Measure stock in the pot; you should have 6 cups. If not, top it up with water until your reach that volume.
Heat the oil in a large, wide pot or Dutch oven set over medium heat. Add the celery, carrot, bell pepper, onion and garlic and cook until softened, about 5 minutes. Mix in the stock, cubed meat and all remaining ingredients, except parsley. Bring to a gently simmer and cook 30 minutes. Season to taste with salt and pepper. Stir in the parsley and serve.
Note: If time allows, you could simmer the meats in the stock the day before finishing off the burgoo. Once lifted out of the stock and cooled to temperature, keep the meat and stock refrigerated until needed. Once the stock is chilled, it will be easier to lift off the solidified fat from its surface.