In a large pot combined corned beef, onion, celery, and pickling spices, add enough water, sprinkle with salt and pepper and bring to a boil. reduce heat to low, cover, and simmer until very tender, 3–3 1/2 hours. Meanwhile, combine dijon mustard and apple cider vinegar in a small bowl and season with salt and pepper to taste. In another bowl, mix mayo, capers, caper brine, and parsley. Season with salt and pepper. Add cabbage and carrots and continue to simmer for 45 minutes to 1 hour more, until cabbage is tender. Remove meat, cabbage, and carrots from pot. Slice corned beef and season with more salt and pepper. Serve with both sauces on the side for dipping.