1 medium onion, trimmed and quartered
Heat the oil in a large Dutch oven over medium-high heat. Add the chicken to the pot, skin-side down, and cook until golden brown, about 5 minutes. Using tongs, turn the chicken and cook until golden brown on the opposite side, about 3 minutes longer. Transfer the chicken to a plate and set aside. Lower the heat to medium and add the andouille. Cook, stirring often, until browned, about 3 minutes. Transfer the sausage to a paper towel-lined plate using a slotted spoon, and set aside.
Lower the heat to medium-low and add the chopped vegetables to the pan. Cook, stirring occasionally, until the vegetables have softened, about 4 minutes. Add the rice, salt, thyme and cayenne; cook, stirring constantly, until the rice is coated with the fat, about 1 minute. Add the diced tomatoes, reserved tomato juice, clam juice, chicken broth, bay leaves and cooked sausage to the pot. Stir to combine. Remove the skin from the chicken pieces and place the chicken on the rice so that the side the skin was just removed from is now facing down. Bring the mixture to a boil, reduce the heat to low, cover and let simmer for 15 minutes. Stir once, keeping the chicken in the same general position, and continue to simmer until the chicken is no longer pink inside, about 10 minutes more. Transfer the chicken to a clean plate or cutting board and set aside. Scatter the shrimp over the rice, cover, and continue to cook until the rice is fully tender and the shrimp are opaque and cooked through, about 5 minutes more.
While the shrimp are cooking, shred the chicken into thin strands. Once the shrimp are finished cooking, discard the bay leaves, stir in the chicken and parsley, and serve immediately.