Bring 2 cups of the stock to a boil in a large soup pot. Add half of the clams, cover and cook just until they open, 5 to 8 minutes; remove the clams to a platter as they open. Return the water to a boil and repeat with the remaining clams. Strain the clam broth through 4 layers of dampened cheesecloth to remove any sand or grit; reserve 4 cups of the broth. Remove the clams from the shells and coarsely chop them.

In a skillet, melt the butter until foamy. Add the flour and cook over moderately low heat, stirring frequently, until the roux is lightly golden, about 15 minutes.

Meanwhile, wipe out the soup pot. Add the bacon and cook over moderate heat, stirring occasionally, until slightly brown and crisp, about 6 minutes. Pour off all but 1 tablespoon of the bacon fat. Add the carrots, onion and celery and cook, stirring occasionally, until softened, about 12 minutes. Add the potatoes, raise the heat to moderate and cook, stirring, until crisp-tender, about 5 minutes.

Add the remaining 1 1/2 cups of stock to the pot along with the reserved clam broth, the cream, milk, pepper and thyme. Simmer over low heat until the potatoes are just tender, about 5 minutes. Add the roux and cook over moderately low heat, stirring, until thickened, about 10 minutes. Add the clams and cook until heated through, about 3 minutes. Ladle the clam chowder into soup bowls and garnish with parsley.