Melt butter in a heavy-bottomed pot over medium-high heat. Cook andouille sausage in the melted butter until completely browned, about 5 minutes. Remove the sausage with a slotted spoon to a plate lined with paper towel.

Add onion, celery, and green bell pepper to the butter remaining in the pot; cook and stir until onion becomes translucent, 5 to 10 minutes. Add garlic; cook and stir together for 1 minute. Stir in tomato paste; cook and stir another 3 to 5 minutes.

Return andouille sausage to the pot along with diced tomatoes, Worcestershire sauce, Cajun seasoning, and black pepper; stir to combine. Pour chicken broth over the sausage mixture; bring to a boil.

Stir in rice; reduce heat to medium-low, cover, and cook until rice is tender, 25 to 30 minutes.

Add shrimp and cod; increase heat to medium-high and bring to a boil. Reduce heat again to medium-low and cook uncovered for another 10 minutes, stirring occasionally.

Season with salt and fold parsley into the jambalaya to serve.