Line a large rimmed tray with newspaper and set out all the seafood, sausage, and vegetables so everything is ready to go.
Fill a very large stockpot with 4 quarts of water. Add the quartered lemons, halved garlic, onion, Cajun seasoning, bay leaves, peppercorns, and salt. Bring to a rolling boil over high heat.
Add the potatoes and sausage. Return to a boil and cook for 12 to 14 minutes, until the potatoes just begin to soften.
Stir in the corn pieces and cook for 8 minutes more.
Add the crab legs and shrimp. Cook for 4 to 6 minutes, just until the shrimp turn pink and curl and the crab is heated through. Do not overcook the seafood.
Turn off the heat and let everything soak in the hot seasoned broth for 10 minutes.
Stir the melted butter with the lemon juice, smoked paprika, and cayenne.
Drain the boil and pile it onto a newspaper lined tray or large platter. Drizzle with the lemon butter, sprinkle with parsley, and serve hot with lemon wedges and crusty bread.