Scrub mussels with a stiff brush under cold running water. Remove any fibrous â€œbeardsâ€� from mussels. Throw away any mussels with open shells.
Soak clams in a bowl, covered with cold water, 15 minutes. Lift out and scrub with a stiff brush under cold running water. Rinse bowl and put clams back in. Cover with fresh cold water, soak 15 minutes, and lift out, leaving any sand in bowl. Keep rinsing clams until no sand remains in bowl.
Bring a large pot of salted water to a boil over high heat. Set a bowl of ice water nearby. Boil shrimp just until pink, about 2 minutes. Using a strainer, transfer to ice water to stop cooking.
Add squid to boiling water. Once pot has returned to boil, cook squid until tentacles curl back and rings are opaque, 45 seconds to 1 minute. Drain well, pat dry with paper towels, and add to bowl of shrimp.
Put 1 tbsp. water (to just cover the bottom) in a 4-qt. saucepan and add cleaned mussels. Cover and cook over high heat until mussels open, 1 to 1 1/2 minutes. Pick out and throw away any mussels that fail to open. Transfer mussels to a clean bowl and let cool 20 minutes.
Add cleaned clams and 1 tbsp. water to pan and steam, covered, over high heat until they open, 1 1/2 minutes. Pick out and throw away any clams that fail to open. Let cool 20 minutes in pan.
Add cooled mussels and clams to bowl of squid and shrimp (leave liquid behind).
In a small bowl, whisk together olive oil, 1/2 cup lemon juice, the garlic, and parsley. Whisk in 1 tsp. salt and several grinds of black pepper.
Pour dressing over seafood and stir gently to bathe with dressing. Let salad marinate at room temperature for 2 hours, tossing occasionally so flavors blend.
Just before serving, taste and season with additional lemon or salt if you like. Serve in shallow bowls, with plenty of the marinade spooned over.
Note: Nutritional analysis is per serving.