Preheat the oven to 375 degrees F. Combine the onion, bread cubes, tomatoes and olives on a rimmed baking sheet Sprinkle with 1/2 teaspoon salt and black pepper to taste. Drizzle with 2 tablespoons of olive oil and toss everything together. Disperse evenly on the baking sheet and bake until the bread just starts to crisp, 12 to 15 minutes.
In a medium bowl, toss the shrimp with 1 tablespoon olive oil and 1/2 teaspoon of salt. Arrange on the baking sheet among the bread mixture. Bake until the shrimp are opaque and just cooked through, 8 to 10 minutes.
Meanwhile, whisk together the remaining 1 tablespoon olive oil, vinegar and anchovy in a small bowl. Stir in the capers and set aside.
Remove the baking sheet from the oven. Add the parsley leaves and dressing and toss to evenly coat. Transfer to a platter and garnish with the chopped parsley. Add Parmesan shavings. Serve immediately.