Put the mayonnaise, mustard, garlic, pickle juice, capers, horseradish, cayenne pepper, paprika, and hot sauce in a bowl and stir to combine. Taste and adjust the seasonings if necessary. Refrigerate until ready to serve.
:::sandwich:::
Mix the buttermilk and salt and pepper to taste together in a medium bowl. Add the oysters and let stand for 5 minutes. Meanwhile, mix the flour, cornmeal, bread crumbs, and Old Bay together in a shallow bowl.
Pour 3 inches of oil into a deep heavy-bottomed pot and heat to 350°F over medium-high heat. Dredge the oysters in the flour mixture, shaking off any excess. Working in batches, fry the oysters until golden brown and crisp, 2 to 3 minutes, and drain on paper towels.
Stuff the rolls with the oysters, spoon some of the sauce over the oysters, and garnish with parsley. Serve with additional sauce and lemon wedges.