In a medium bowl, stir together the horseradish, applesauce, vinegar, mustard, and mayonnaise. Season with salt and pepper. DO AHEAD: The sauce can be made and stored, in an airtight container in the refrigerator, up to 3 days.

Working in batches, use the larger side of a box grater or a food processor fitted with a grater attachment to coarsely grate the potatoes, Jerusalem artichokes, and onion. As they're grated, transfer the vegetables to the towel-lined bowl. Gather the corners of the towel and wring as much excess liquid as possible from the vegetables.

Transfer the wrung vegetables to a dry mixing bowl. Add the egg and parsley and stir to combine. While stirring the mixture, gradually add the flour, stirring well to incorporate.

In a large sauté pan over moderately high heat, heat ¼” inch of oil until hot but not smoking. If necessary, add additional flour, 1 tablespoon at a time, but try to add as little flour as possible to create light latkes.

Season with salt and pepper and fry until golden brown, about 2 minutes. Flip the latkes and continue frying until golden brown, about 2 minutes. As they finish cooking, transfer the latkes to the paper-towel-lined baking sheet.