Rub the chicken thighs all over with the cajun seasoning. Cook until the spices begin to blacken and the thighs are cooked through. Transfer the chicken thighs to a plate and set aside.
Add the garlic, onion, celery, bell pepper, and andouille sausage to the pan and saute until the vegetables have softened slightly, about 5 minutes. Stir in the rice and chicken broth.
Bring the mixture to a boil, then reduce to a simmer and cover. Cook for 20 minutes, or until the liquid has absorbed. Return the chicken to the pan and cover again until the chicken is heated through and the rice is fully cooked, about 5 additional minutes. Season with Kosher salt, to taste.
Sprinkle with fresh parsley and serve warm with lime wedges.