Fill a large bowl halfway with ice and add water to cover; set aside. Cut one lemon into wedges.
In large pot combine 6 cups water, salt, sugar, onion, and 5 sprigs parsley. Halve and juice remaining lemon and add to pot. Bring to a boil, then turn off heat and add shrimp to pot.
Let sit until shrimp are pink and cooked through, about 3 minutes. Remove shrimp from the pot with a slotted spoon (or spider) into the bowl of ice.
Make the cocktail sauce: In a small bowl, mix all ingredients until fully combined.
Serve shrimp with lemon wedges, cocktail sauce, and garnish with parsley.
