Heat olive oil in a large skillet or Dutch oven over medium heat. Add the diced onion, bell pepper, and celery. Sauté for about 5 minutes, or until the vegetables are softened.
Add the minced garlic to the skillet and cook for an additional minute, stirring frequently to prevent burning.
Slice the gluten-free sausage and add it to the skillet. Cook until browned, about 5-7 minutes.
Stir in the rice, Cajun seasoning, dried thyme, black pepper, and salt. Mix well to coat the rice with the flavor of the sausage and vegetables.
Pour in the chicken broth and bring the mixture to a gentle boil. Once boiling, reduce the heat to low, cover, and let simmer for about 20 minutes, or until the rice is tender and has absorbed the liquid.
Remove from heat, fluff the rice with a fork, and let it rest for 5 minutes before serving.
Serve hot, garnished with sliced green onions and chopped parsley.