Cook the bacon in a wide heavy-duty 4- to 5-quart pot or Dutch oven over medium heat, stirring occasionally, until just beginning to turn crisp and golden, about 4 minutes. Remove from the heat. Pour off and discard the bacon fat, leaving the bacon in the pot. Add the butter and onion and cook over low heat, covered, stirring occasionally, until the onion is softened but not colored, about 8 minutes. Add the celery and cook, uncovered, stirring occasionally, until just softened, about 5 minutes. Add the clam broth, potatoes, bay leaves, thyme, and 1 tsp. pepper. Bring to a boil over high heat. Lower the heat to maintain a simmer and cook, partially covered, until the potatoes are tender, 10 to 12 minutes. Discard the bay leaves.
Purée 1 cup of the soup solids with just enough liquid to cover in a food processor or blender, and add it back to the soup. Add the cream and bring to a boil.
Remove the soup from the heat, wait until it stops simmering (this may take a minute if you’re using a Dutch oven), and stir in the clams and parsley. Season to taste with salt and pepper.