Cook bacon in a Dutch oven over medium-high heat until crisp. Remove bacon from pan, reserving 2 teaspoons drippings in pan. Crumble bacon; set aside.
Place pan over medium-high heat. Add onion and celery to drippings in pan; sauté 5 minutes or until tender. Stir in broth and parsnip; bring to a boil. Cover, reduce heat, and simmer 5 minutes. Stir in potato; cover and simmer 19 minutes or until vegetables are tender. Remove from heat; stir in milk, salt, and pepper.
Spoon 3 cups soup into a large bowl; mash vegetables with a potato masher, and return soup to pan. Stir in parsley. Ladle soup into bowls, and sprinkle each with crumbled bacon.