Step 1

In a large bowl, whisk together lemon zest, juice, 1/2 teaspoon sugar, 1 teaspoon salt, 1/2 teaspoon pepper, and garlic. Whisk in oil and chopped parsley.

Step 2

Meanwhile, in a large pot, bring 4 quarts water, 6 tablespoons sugar, and 1/4 cup salt to a boil. Add squid and blanch until just tender, 30 seconds. Using a slotted spoon or a spider, transfer squid to an ice bath to stop cooking. Once cool, drain squid and lay on paper towels to dry; transfer to bowl with dressing.

Step 3

Return water to a boil. Add shrimp, stir once, cover, and remove from heat. Let stand 3 minutes; using a slotted spoon, remove to a plate to cool slightly; pat dry and transfer to bowl with squid. Return water to a boil and add mussels, reduce heat to medium, cover, and simmer until mussels open, 3 to 5 minutes (discard any unopened mussels). Drain and transfer to a plate to cool. Remove all but 10 mussels from their shells and add to bowl with seafood. Reserve mussels in shell for serving.

Step 4

Toss dressed seafood with fennel, bell pepper, celery, crab, and parsley leaves. Season with salt and pepper. Serve immediately, with reserved mussels and lemon wedges, or store, covered, in the refrigerator, for up to 8 hours.