Rinse the clams several times under cold running water. Transfer to a large pot and add 3 1/2 cups water. Bring to a simmer over medium heat, then cover and cook until the clams open, about 8 minutes. Remove the clams with a slotted spoon and transfer to a deep bowl as they open; cover with plastic wrap so they don't dry out. Strain the broth through a fine-mesh sieve into a large bowl. Wipe out the pot, then pour the strained broth back into the pot.
Meanwhile, put the bacon in a large skillet and cover with about 2 inches of water. Bring to a simmer over medium heat and cook 10 minutes; drain, then return the bacon to the skillet. Add 1 tablespoon butter and cook until the bacon starts to brown, about 5 minutes. Transfer the bacon to the pot with the broth. Wipe out the skillet. Add the remaining 1 tablespoon butter, the onion and celery to the skillet and cook over medium heat until the onion is translucent, about 6 minutes.
Add the onion and celery mixture to the broth along with the potatoes. Bring to a simmer over medium heat and cook until the potatoes are tender, 10 to 12 minutes. Remove from the heat. Stir in the Worcestershire sauce and 1/2 teaspoon white pepper.
While the potatoes are cooking, remove the clams from their shells and roughly chop.
Add the clams to the soup and cook until heated through, about 1 minute. Ladle the soup into bowls and top with chopped parsley.