In food processor, pulse onion, celery, red pepper, and garlic until chopped fine, about six 1-second pulses, scraping down sides of bowl once or twice. Do not overprocess; vegetables should not be pureed.
Heat oil in large heavy-bottomed Dutch oven over medium-high heat until shimmering but not smoking, about 2 minutes. Add chicken, skin-side down, and cook until golden brown, about 5 minutes. Using tongs, turn chicken and cook until golden brown on second side, about 3 minutes longer.
Transfer chicken to plate and set aside. Reduce heat to medium and add andouille; cook, stirring frequently, until browned, about 3 minutes. Using slotted spoon, transfer sausage to paper towel–lined plate and set aside.
Reduce heat to medium-low, add vegetables, and cook, stirring occasionally and scraping bottom of pot with wooden spoon, until vegetables have softened, about 4 minutes.
Add rice, salt, thyme, and cayenne; cook, stirring frequently, until rice is coated with fat, about 1 minute. Add tomatoes, reserved tomato juice, clam juice, chicken broth, bay leaves, and browned sausage to pot; stir to combine.
Remove and discard skin from chicken; place chicken, skinned-side down, on rice. Bring to boil, reduce heat to low, cover, and simmer for 15 minutes. Stir once, keeping chicken on top, skinned-side down. Replace cover and continue to simmer until chicken is no longer pink when cut into with paring knife, about 10 minutes more;.
Transfer chicken to clean plate and set aside. Scatter shrimp over rice, cover, and continue to cook until rice is fully tender and shrimp are opaque and cooked through, about 5 minutes more.
While shrimp are cooking, shred chicken. When shrimp are cooked, discard bay leaves; off heat, stir in parsley and shredded chicken, and serve immediately.