Heat olive oil in a large heavy-bottomed pot or Dutch oven over medium-high heat.
Add the chicken pieces and cook until browned on all sides, about 5 minutes. Remove chicken and set aside.
Add the sliced smoked turkey sausage to the pot and cook until browned, about 3 minutes. Remove sausage and set aside with the chicken.
Add diced onion, green bell pepper, and celery to the pot. Cook, stirring occasionally, until vegetables are softened, about 5 minutes.
Add minced garlic and cook for 1 minute until fragrant.
Stir in the diced tomatoes with their juice, smoked paprika, dried thyme, dried oregano, cayenne pepper, black pepper, salt, and bay leaves. Cook for 2 minutes to combine flavors.
Add the rice to the pot and stir well to coat the rice with the tomato and spice mixture.
Return the browned chicken and sausage to the pot and pour in the chicken broth. Stir to combine.
Bring the mixture to a boil, then reduce heat to low. Cover the pot and simmer gently for 25 to 30 minutes, or until the rice is tender and the liquid is absorbed.
Remove the pot from heat and let it sit, covered, for 5 minutes.
Discard the bay leaves and fluff the jambalaya with a fork.
Stir in the chopped fresh parsley before serving.