Begin by preparing all seafood. Rinse the shrimp, scallops, and mussels under cold water and pat dry. Ensure the crab legs are thawed if frozen.
In a large skillet or on a grill, heat olive oil over medium-high heat. Season the shrimp and scallops with salt and pepper.
Place the shrimp and scallops in the skillet and cook for about 2-3 minutes on each side until they are opaque and cooked through. Remove and set aside.
In the same skillet, add the mussels and cover. Cook for about 5-7 minutes until they open up, discarding any that remain closed.
For the crab legs, if they're precooked, just heat them in boiling water for about 5 minutes. For raw crab legs, steam them for 10-15 minutes until fully cooked.
Arrange all seafood on a large serving platter over a bed of ice, if desired, for presentation. Garnish with lemon wedges and fresh parsley.
In small bowls, mix together the ingredients for dipping sauces. Serve alongside the seafood platter.