Heat oven to 350 degrees. Put potatoes in a large pot and cover with cold water; season with salt. Bring potatoes to a boil, then cook until they are just tender but not falling apart, about 5 minutes. Drain and set aside.

Cook bacon in a large skillet over medium heat, stirring often, about 8 minutes; remove with a slotted spoon and toss with the potatoes. Measure the bacon fat left in the skillet, adding olive oil if necessary to equal 1/4 cup, and return to the skillet.

Set skillet over medium heat and add celery and onion. Cook, stirring, 1 minute. Stir in flour until incorporated. Slowly add 1 1/3 cups water and the vinegar, stirring constantly, until mixture thickens enough to coat the back of a spoon, 2 to 3 minutes.

Remove skillet from heat and stir in sugar, 1 1/2 teaspoons salt, celery seed, mustard, pepper and parsley. Pour over potatoes and bacon and gently toss to coat.

Transfer potatoes to a 3-quart baking dish; cover with foil and bake for 35 minutes. Let stand 10 minutes before serving.