Place lasagna noodles in a large bowl of warm water and let soak, stirring occasionally, until pliable, at least 10 minutes.
Meanwhile, combine clam juice, wine and water in a large deep skillet. Bring to a simmer. Reduce heat to medium-low and add shrimp. Poach until pink, about 40 seconds. Remove with a slotted spoon and transfer to a colander over a bowl.
Add sole and salmon to poaching liquid. Poach until just opaque, about 1 minute. Remove with a slotted spoon and place in colander with shrimp. Let drain. Strain poaching liquid through a fine sieve and set aside. Wipe pan dry.
Heat 1 1/2 teaspoons oil in the pan over medium-high heat. Add shallots, anchovies and garlic and cook, mashing anchovies into a paste, for about 1 minute. Add mushrooms and sauté until browned, about 2 minutes. Add basil and season with salt and pepper. Transfer to colander with the seafood. Add any accumulated juices to the reserved poaching liquid. Measure out 3 1/2 cups hot liquid, adding water if necessary.
Heat the remaining 2 tablespoons oil in a large heavy saucepan over medium heat. Add flour and cook, whisking constantly, for 1 minute. Add the reserved poaching liquid and bring to a simmer, whisking, until smooth and thickened to the consistency of heavy cream, 5 to 6 minutes. Add lemon juice and season with salt, pepper and cayenne. Reserve 1 1/2 cups sauce; gently mix seafood mixture into remaining sauce.
Preheat oven to 350°F. Coat a 9-by-13-1/2-inch baking dish with cooking spray.
Spoon 1/2 cup reserved sauce into the prepared dish. Drain the noodles and blot dry. Alternate 4 layers of noodles and 3 layers of the seafood mixture in dish, starting and ending with noodles. Spread remaining 1 cup sauce over the top, coating the noodles completely.
Cover the lasagna with foil and bake for 25 minutes. Uncover, sprinkle with breadcrumbs and Parmesan and bake until golden and bubbly, 15 to 20 minutes more. Let stand for 10 minutes. Garnish with parsley and serve.