In a 6- to 8-quart pan, combine potatoes and 3 quarts water. Cover and bring to a boil over high heat. Reduce heat and simmer until potatoes are tender when pierced, 20 to 30 minutes. Drain well and let stand until cool enough to touch, 15 to 25 minutes.
Meanwhile, in a small bowl, soak mustard seeds in about 1/2 cup hot water until soft, about 5 minutes. Drain.
In a large bowl, mix mustard seeds, chopped pickles, pickle juice, mayonnaise, and vinegar. Peel warm potatoes, cut into about 3/4-inch cubes, and drop into dressing. Add bell peppers; mix gently. Let cool to room temperature, at least 15 minutes. Add 1/2 cup parsley and salt and pepper to taste; mix gently. Scrape into a serving bowl and sprinkle with remaining parsley (see notes).