Combine the carrots, celery, cabbage, pancetta, beans, tomato paste, parmesan rind, 1 teaspoon salt, 1/2 teaspoon pepper and 7 cups water in a 5-to-6-quart slow cooker. Cover and cook on low, 7 to 8 hours. Leave 1 slice of bread at room temperature to dry out slightly.
Before serving, add the stale bread to the slow cooker. Cover and continue to cook, about 10 minutes. Stir the soup well, then add the grated parmesan and parsley. Season with salt and pepper.
Meanwhile, preheat the broiler. Spread the remaining 4 bread slices on a baking sheet and drizzle with olive oil. Broil until golden, 1 to 2 minutes per side. Place a slice in each bowl.
Heat 3 tablespoons olive oil and the garlic in a medium skillet over medium-high heat. Cook, swirling the pan frequently, until the garlic is golden brown, about 3 minutes. Stir the garlic and oil into the soup. Ladle the soup over the bread in the bowls.