Melt butter in a stockpot over medium-high heat; cook and stir salt pork, celery, onion, and garlic in melted butter until softened, about 7 minutes. Add potatoes, clam juice, tarragon, celery salt, and bay leaves to the stockpot. Bring the mixture to a boil, reduce heat to medium, and place a cover on the pot. Cook at a simmer until potatoes are tender, about 35 minutes.
Stir clams into the soup and simmer uncovered for 5 minutes. Add Worcestershire sauce, heavy cream, dill, salt, and pepper; stir and cook another 7 minutes. Remove the pot from heat. Remove and discard bay leaves. Garnish with parsley to serve.