Heat the olive oil in a large skillet over medium-high. Brown the andouille 3–4 minutes until the edges crisp. Transfer to a 6-quart slow cooker.
Sear the chicken in the same skillet for 4–5 minutes, until colored on the outside. Add to the slow cooker.
Add the onion, bell pepper, and celery to the skillet and cook 4–5 minutes until the onion is translucent. Stir in the garlic and cook 1 minute more. Scrape everything into the slow cooker.
Add the diced tomatoes with juice, tomato paste, Cajun seasoning, smoked paprika, thyme, bay leaves, and ½ cup of the broth. Stir to combine.
Cover and cook on LOW for 5–6 hours or HIGH for 3 hours.
Turn to HIGH. Stir in the rice and the remaining 2 cups of broth, submerging the rice. Cover and cook 40–45 minutes, until the rice is tender.
Stir in the shrimp. Cover and cook 15–20 minutes on HIGH, until the shrimp are pink and opaque.
Remove the bay leaves. Stir in the green onions and parsley. Taste and adjust seasoning before serving.