Place octopus in a large pot with bay leaves, peppercorns, celery, and carrot. Cover completely with cold water, and bring to a rapid boil. Drop heat, and allow to simmer for 45-90 minutes or until a sharp knife can pierce through it easily.
Drain octopus, reserving boiling liquid. Allow octopus to slightly cool. Cut Tentacles off head, and cut into one centimeter rounds. Scoop out the core of the head, and discard. Cut head into thin pieces.
Bring remaining boiling liquid to a simmer, and add potatoes and enough water (if necessary) to cover and season with salt. Cook until fork tender. Drain potatoes, cool, peel, and cut into bite size cubes.
Whisk together lemon juice, olive oil, and salt and pepper to taste. Pour over octopus and potatoes, along with capers and parsley. Toss gently to coat.
Serve slightly warm, or at room temperature with crusty bread.