Heat oil in a large heavy pot or Dutch oven over medium-high heat.
Season chicken with salt, pepper and 1 tsp Creole seasoning; brown in batches until golden; remove and set aside.
Add sliced sausage to the pot and brown lightly; remove with the chicken.
Reduce heat to medium, add onion, bell pepper and celery; sauté until softened, about 5–7 minutes.
Stir in garlic and remaining Creole seasoning and cook 30 seconds until fragrant.
Add rice and stir to coat the grains in oil and vegetables for 1–2 minutes.
Pour in diced tomatoes and chicken stock; add thyme, bay leaves, and return chicken and sausage to the pot.
Bring to a simmer, cover, and cook gently for 18–20 minutes until rice is nearly tender.
Stir in shrimp, cover, and cook 3–5 minutes until shrimp are pink and cooked through.
Remove bay leaves, adjust seasoning with salt and pepper, stir in parsley or green onions, and let rest 5 minutes before serving.