Heat the oil over medium heat in a large, heavy soup pot or Dutch oven. Sprinkle in the flour and blend with a whisk until well combined. Continue cooking, whisking frequently, until the mixture develops a rich, brown color, 8 to 10 minutes.
Add the andouille, onion, green pepper and celery. Continue cooking, stirring frequently, until the onion is soft and translucent, 6 to 8 minutes longer. Add the diced tomatoes and 2 cups of chicken broth. Stir until smooth and well blended and season to taste with salt and pepper.
Cover and cook over medium-low heat for 35 to 40 minutes, stirring occasionally. Increase the heat to medium, add the okra and cook until tender, 10 to 12 minutes more.
Note: If you're looking for a more soup-like consistency, feel free to add more broth as the gumbo simmers.
Stir in the shrimp and continue cooking until pink and opaque, 4 to 5 minutes longer. Taste and adjust the seasoning if needed.
To serve, ladle the gumbo into shallow bowls, top with a serving of steamed rice and garnish with parsley.