In a large saucepan over medium heat, cook the bacon until crisp, 6 to 8 minutes. Transfer to a paper towel-lined plate. Let cool, then break into pieces.
Return the saucepan to medium heat, add the carrot to the pan drippings, and cook for 5 minutes.
Add the celery, onion, and garlic, stirring well and cooking for 3 minutes more.
Add the beans and 2 cups water. Bring to a boil, cover, and reduce heat to low. Let simmer for 10 minutes.
Uncover and, with a potato masher or large slotted spoon, partially mash the bean mixture until it thickens slightly.
Stir in the parsley, 1 ¼ teaspoons salt, ½ teaspoon pepper, and bacon.