In a large Dutch oven, heat 2 tablespoons of the olive oil over medium high heat.
Season shrimp with Creole seasoning and sauté until almost cooked through, about 3 minutes. Using a slotted spoon, remove shrimp and set aside until later.
Season chicken pieces with Creole seasoning. Add remaining olive oil to Dutch oven and when hot, add chicken pieces and sauté until browned on both sides, about 8 minutes. Remove and set aside.
Add sausage and cook until browned. Add onion, celery, bell pepper and saute until wilted--about 5 minutes. Add garlic and saute another 1 to 2 minutes. Add bay leaves, cayenne and thyme and cook additional couple minutes--stirring.
Add tomatoes and water (or chicken stock) and return chicken pieces to pot. Season with salt and pepper and simmer, covered, for 20 minutes.
Add rice to pot, stir well and return to a boil. Reduce heat to low, cover pot, and cook for 15 minutes.
Add shrimp, green onions and parsley to Dutch oven, mixing carefully, and continue to cook, covered, for another 10 minutes.
Remove from heat and let sit 10 minutes before serving.
Note: Chopped onion, celery and bell pepper (2:1:1) make up the "holy trinity" of Southern cooking. Add 1/2-1 cup chopped red or yellow bell pepper and saute with "holy trinity" for color and sweetness.