Put the bacon in a wide heavy-duty 4- to 5-quart pot or Dutch oven and cook 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 sweet and tender, about 10 minutes. The onion shouldn’t color. Add the celery and garlic and cook, uncovered, stirring occasionally, until all the vegetables are tender, 5 to 7 minutes more. Add the clam broth, potatoes, thyme, bay leaves, and 1/4 tsp. pepper. Bring to a boil over high heat. Lower the heat to maintain a simmer and cook until the potatoes are tender, about 15 minutes.
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 chopped parsley. Season to taste with black pepper, if desired.