Step 1: Heat the vegetable oil in a large Dutch oven or heavy-bottomed pot over medium-high heat. Add the sliced andouille sausage and chicken pieces. Cook until browned, about 8-10 minutes. Remove the meats with a slotted spoon and set aside.
Step 2: In the same pot, add the onion, bell pepper, and celery (the Holy Trinity). Sauté for about 5-7 minutes until the vegetables are softened and the onions are translucent.
Step 3: Stir in the minced garlic and cook for another minute until fragrant. Add the uncooked rice to the pot and stir for 1-2 minutes to lightly toast the grains.
Step 4: Pour in the chicken broth and add the Cajun seasoning, thyme, and smoked paprika. Stir to combine, ensuring you scrape up any browned bits from the bottom of the pot. Return the sausage and chicken to the pot.
Step 5: Bring the mixture to a boil, then reduce the heat to low. Cover the pot with a tight-fitting lid and simmer for 20-25 minutes, or until the rice is tender and the liquid is absorbed.
Step 6: Gently fold in the shrimp. Replace the lid and cook for another 5 minutes, or until the shrimp are pink and opaque. Turn off the heat and let the jambalaya rest for 5 minutes before serving. Garnish with green onions and parsley.