In a large, heavy-bottomed saucepan, stir together the cream, milk, garlic, bay leaves, thyme, anchovies, 1 1/4 teaspoons salt, and 1/4 teaspoon pepper until the salt dissolves.
Add the cod and bring to a simmer over medium-high heat. Reduce the heat to medium and continue to simmer until the fish flakes apart easily, about 3 minutes. Using a slotted spoon, transfer the fish to a bowl and reserve.
Add the potatoes to the cream mixture and bring the liquid to a simmer over medium-high heat. Reduce the heat to low, cover, and cook, stirring often, until the potatoes are tender, 10 to 15 minutes.
Remove the pot from the heat and discard the bay leaves and thyme. Using a potato masher, mash the potatoes and the cream mixture until smooth.
Gently fold 3 tablespoons parsley and the reserved cod into the potatoes, trying to keep the flakes of cod intact if desired. Adjust the seasonings with salt and pepper to taste.