For the shrimp cocktail: Add the parsley stems, peppercorns, garlic, lemon, shallot, 2 teaspoons salt and 3 quarts water to a large stockpot and stir to combine. Bring to a boil over high heat. Prepare an ice water bath.
Add the shrimp to the boiling water and cook just until opaque, 1 to 2 minutes. Remove the shrimp and shock in the ice water bath for 5 minutes. Drain the shrimp and refrigerate until ready to serve. Reserve the parsley leaves for the peppercorn remoulade sauce.
For the peppercorn remoulade sauce: Combine the mayonnaise, green peppercorns, brine and sugar in a small bowl. Chop the reserved parsley leaves and stir into the remoulade. Store in an airtight container in the refrigerator until ready to serve.
For the "bloody" cocktail sauce: Combine the ketchup, horseradish, celery seed and Worcestershire sauce in a small bowl. Store in an airtight container in the refrigerator until ready to serve. This can be prepared up to 2 weeks in advance.
Serve the chilled shrimp over crushed ice with the dipping sauces alongside.