Heat oil in a large dutch oven equipped with a tight-fitting lid. Add garlic and spices and toast about a minute until fragrant. Add Andouille sausage and brown for about 5 minutes, stirring often. Remove sausage with a slotted spoon and set aside. Add celery, onion, and bell pepper to the hot, spiced drippings from the sausage. Cook, stirring often, until the onions begin to sweat (about 5 minutes.) Scrape the bottom of the pot to incorporate all the flavor from browning the sausage into the dish. 

Return the sausage to the pot, add the chicken, and then add tomatoes, broth, and rice. Stir thoroughly and bring to a boil over high heat. Once boiling, reduce heat to low, cover, and simmer until the rice is tender. Stir occasionally. This should take about 20 minutes. 

When rice is tender and most of liquid is absorbed, take a moment to taste the dish. Adjust salt to taste. (So far the dish has no added salt. If your Andouille and chicken are salty, it may not need any.) Also assess the heat of the dish. If the spice heat from the Andouille and Creole Seasoning is too mild, add 1/4 tsp - 1 tsp cayenne pepper to liven it up!

Add shrimp, cover, and cook another 5 minutes on low--until the shrimp have just turned pink. Add parsley, and serve immediately!