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.
Cook and stir onion, celery, and green bell pepper in the butter remaining in the pot until the onion becomes translucent, 5 to 10 minutes.
Stir garlic into the onion mixture; cook and stir together for 1 minute.
Stir tomato paste through the vegetable mixture; 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 rice into the mixture, place a cover on the pot, reduce heat to medium-low, and cook undisturbed until the rice is tender, 25 to 30 minutes.
Add shrimp and cod to the rice mixture. Increase heat to medium-high and cook until the mixture comes to a boil; reduce heat again to medium-low and cook uncovered another 10 minutes, stirring occasionally.
Season with salt and fold parsley into the jambalaya to serve.