In a large stockpot or Dutch oven over medium heat, cook the bacon until crisp, about 10 minutes. Drain the bacon on paper towels and set aside.

Pour off all but about 2 tablespoons of the bacon fat from the stockpot. Add the onions and celery and sauté over medium heat, stirring occasionally, until they begin to soften, about 5 minutes. Add the potatoes and just enough seafood stock or clam juice to cover them, adding some cold water if necessary. Raise the heat and bring the liquid to a boil; then, reduce the heat to medium-low, cover the pot, and cook until the potatoes are almost tender enough to pierce with a fork, about 15 minutes.

Uncover the pot and carefully lay the fish fillets on top of the vegetables. Cover the pot again and cook the fish until barely opaque, about 5 minutes. Add the cream and thyme and gently stir, trying not to break the fish fillets into too many pieces. Remove the pot from the heat and leave it to cool to room temperature, then refrigerate for at least 3 to 4 hours or, for an even better flavor, overnight.

Before serving, warm the soup over medium-low heat, stirring in the 2 tablespoons butter. Ladle into heated bowls and garnish with bacon bits and parsley.