In a large bowl, whisk the anchovies with garlic until pasty. Whisk in the lemon juice and vinegar. Gradually whisk in the oil. Season with salt and pepper to taste. Set aside.
Cover and heat a medium pot of water to a boil over high heat. Season generously with kosher salt. Reduce heat to a simmer. Add the squid and simmer for 2–3 minutes, until white and opaque. With a slotted spoon, transfer the squid to a colander set in the sink and rinse with cold water to stop the cooking. Drain well and add to the bowl with the dressing. To the same pot of simmering water, add the scallops and cook for 2–3 minutes, until opaque. Transfer to the colander and rinse with cold water. Drain well and add to the serving bowl.
To the same pot of simmering water, add the shrimp. Cook for 2–3 minutes, until cooked through. The shrimp should look opaque with pink and bright red-orange accents. Transfer shrimp to the colander and rinse with cold water. Drain shrimp well and add to the serving bowl.
To the same pot of simmering water, add the cod. Cook for 4–5 minutes, until firm and cooked through. With a slotted spoon, transfer to a plate and set aside (you’ll add the cod to the salad last). Carefully discard most of the water in the pot, leaving about 2 inches. Add the mussels to the pot. Cover and cook for 3–5 minutes, until all the shells have opened (discard any that do not open at all). Transfer mussels to a colander to cool. Remove mussels from the shells and add to the bowl with the squid, scallops and shrimp.
To the bowl with seafood, add the celery, clams, and parsley. Toss until well combined. Season with salt and pepper to taste. Add the cod and gently toss to combine. Refrigerate, covered, for at least 2 hours or up to 6 hours before serving. Garnish with lemon slices.