In a large saucepan over medium-low heat, gently melt the butter until shimmering. Add the onion, carrot, and celery, and sauté for 3 minutes until tender but not browned.
Stir in the flour until the mixture bubbles and pulls away from the side of the pan. Cook for an additional 2 minutes, then remove from heat. Add the fish stock and milk, making sure to scrape the bottom of the pan to prevent lumps in the soup.
Place the pan back on the heat, lower it to a gentle simmer, and simmer for 25 minutes. Stir in the prawns, mussels, salmon, and parsley and cook for an additional 4-5 minutes until the seafood is just cooked. Season with salt and pepper to taste.
Divide the soup evenly among bowls and serve right away.