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Bring a medium pot of salted water to a boil. Add the potatoes and cook until tender all the way through (test them with a skewer). Start testing after about 5 minutes for small potatoes and 8 minutes for medium. Drain the potatoes and let them cool for 10 minutes.
Scrape the skins off the potatoes with a paring knife and dice them into a large mixing bowl. Add the mayonnaise and mustard and stir till evenly coated. Add more mayonnaise if you like. Stir in the onion, celery and parsley and season with salt and pepper. Cover and refrigerate until ready to serve.
Prepare grill or preheat grill pan.
Grill hot dogs. Set a dog inside each bun and serve with potato salad alongside.