Adjust an oven rack to the lower middle position and preheat to 350°F (175°C).

Season the chicken all over with the salt and pepper.

Heat the oil in a large Dutch oven or heavy oven-proof soup pot over medium-high heat until it shimmers. Brown the chicken on both sides, flipping once, about 8 minutes total; transfer to a large bowl.

Add the sausage to the fat left in the pot and cook, stirring frequently, until lightly browned, about 5 minutes. Add to the bowl with the chicken.

Add the onion, celery, and bell pepper to the fat in the pot and cook, stirring frequently, until softened, about 5 minutes. Add the garlic and cook 30 seconds more (do not brown). Stir in the tomato paste and cook for 1 minute more.

Add the chicken broth, clam juice, diced tomatoes, thyme, and cayenne pepper. Using a wooden spoon, scrape up any brown bits from the bottom of the pot. Add the chicken and sausage back to the pot, along with any accumulated juices, and bring to a simmer. Cover and cook over medium-low heat for 20 minutes.

Add the rice and stir to combine. Cover the pot, place in the oven, and cook until the rice is tender and all of the liquid has been absorbed, about 25 minutes, gently stirring the rice once halfway through cooking.

Transfer the chicken to a cutting board. Stir the shrimp into the rice mixture, cover, and continue to cook in the oven until the shrimp are opaque throughout, 5 to 7 minutes. Using two spoons, pull the chicken off the bones; discard skin and bones.

Gently stir the pulled chicken and parsley (if using) into the rice. Taste and adjust seasoning if needed. Cover and let sit until the chicken is heated through, about 5 minutes. Serve.