Heat a 6-quart pot over medium-high heat. Add the oil. Season the chicken with salt and pepper and fry until golden brown on both sides and cooked through, 20 to 25 minutes. Remove and set aside, leaving the oil in the pot.
To make the roux, reduce the heat to medium and stir the flour into the hot oil, making a thick paste. Cook, stirring, until the roux is the color of dark peanut butter, about 20 minutes. Watch closely and stir constantly because roux are easily burned.
Stir in the garlic, celery, onion, bell pepper and bay leaves and let them sweat until the vegetables are soft. Add the andouille, chicken stock and salt and pepper to taste and bring to a boil. Reduce the heat to low and simmer for 1-1/2 hours, stirring occasionally to keep the roux from sticking to the bottom of the pot and burning.
While the gumbo simmers, pull the meat from the chicken. Stir it into the gumbo the last 30 minutes or so of cooking, along with the shrimp.
Remove the gumbo from the heat, discard the bay leaves (if used) and stir in the green onions and parsley. Serve over rice.