In a large saucepan, place the potatoes and cover with cold water by 1 inch. Stir in 2 tablespoons salt. Bring to a boil, lower the heat, and simmer until just tender, about 30 minutes. Drain, transfer to a large bowl, and cover with plastic wrap to keep warm.
Meanwhile in a small saucepan, over medium-high heat, add the bacon and cook, stirring frequently, until crispy and its fat has been released, about 3 minutes. Lower the heat to medium, add the onion and cook, stirring, until translucent, about 6 minutes. Stir in the flour and cook for 45 seconds more. Add the remaining 2 teaspoons salt, broth, vinegar, mustard, and sugar. Bring to a boil while whisking constantly. Remove from the heat.
Using your hands, rub the skins off the potatoes. Cut the potatoes in half lengthwise, then cut crosswise into 1/4-inch slices. Return the potatoes to the large bowl. Pour over the vinegar mixture and toss to coat the potatoes. Toss in the parsley and season with pepper to taste. Transfer to a serving platter and serve immediately.
This potato salad is at its best served immediately or shortly afterwards at room temperature.
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