Cut 2 lemons into wedges; set half the wedges aside for serving.
In a large pot or Dutch oven, combine 16 cups water, the reserved lemon wedges for cooking (from step 1), the peeled red onion cut into 4 wedges, 4 cloves minced garlic, and 1 tablespoon Old Bay seasoning. Stir and bring to a rolling boil over medium-high heat.
Add the 1½ lb red baby potatoes (halved or quartered as noted) and 1 tablespoon salt. Return to a boil, then cook 10–12 minutes, until the potatoes are almost tender when pierced with a fork.
Add the 4 ears corn (each cut into 4–5 pieces) and the 14–16 oz sliced smoked andouille sausage. Cook 4–5 minutes more, until the corn is heated through and the sausage is warmed.
Remove the pot from the heat. Immediately add the 1½ lb raw large shrimp (peeled and deveined, tails left on) to the hot liquid and stir gently. Cook 1–2 minutes, just until the shrimp turn opaque. Do not overcook.
Drain the mixture into a large colander, reserving ½ cup of the cooking liquid.
Arrange the potatoes, corn, sausage, and shrimp on a large platter or sheet pan.
Drizzle with the ¼ cup melted unsalted butter and, if desired, a few tablespoons of the reserved cooking liquid. Squeeze the remaining lemon wedges (from step 1) over the seafood and vegetables.
Finish with ground black pepper to taste and sprinkle with the 2 tablespoons chopped parsley. Serve immediately.