If you prefer your pasta al dente, add it after the chicken has cooked. Slideshow: More Crock Pot and Slow Cooker Recipes
Heat a slow cooker on high until hot. Stir in the butter until melted, then stir in the onion, carrots, celery, 1/2 teaspoon salt and 1/4 teaspoon pepper and cook until golden, 6 to 8 minutes. Stir in the flour and cook 2 minutes. Reduce the heat to low and stir in the chicken, stock, and pasta. Cook until the chicken is very tender, 6 to 8 hours.
Remove the chicken and cool to warm. Remove the meat, discarding the bones. Return the chicken to the slow cooker. Season the soup with salt and pepper to taste, then serve sprinkled with the parsley.