Heat the olive oil in a large pot over medium heat. Add the sliced Andouille sausage and cook until lightly browned on both sides, about 5 minutes. Remove the sausage and set it aside.
Add the onion, celery, and bell peppers to the same pot. Cook for 4 to 5 minutes until the vegetables begin to soften. Stir in the garlic and Cajun seasoning and cook for 1 minute more.
Return the sausage to the pot. Add the rice, drained diced tomatoes, and chicken broth. Stir once to combine, then bring the mixture to a boil.
Reduce the heat to low, cover the pot, and simmer for 20 minutes until the rice is tender and the liquid is absorbed. Remove from heat and let it rest for 5 minutes before fluffing with a fork.
Spoon into bowls and top with chopped parsley.