Roasting the carrots until they're slightly charred really brings out the sweetness of the vegetable.
Preheat oven to 375 degrees. In a large bowl, toss bread cubes with 3 tablespoons oil and the Parmesan. In another large bowl, toss carrots with 2 tablespoons oil; season with salt and pepper.
Lay carrots and bread on a baking sheet in 1 layer, with carrots on 1 side and bread on the other, and bake for 25 minutes. Remove croutons when crisp and golden, about 25 minutes. Spread out carrots, and cook until caramelized and slightly charred, about 25 minutes more.
Assemble the salad: Toss parsley, carrots, lemon juice, and the remaining 2 tablespoons oil. Add croutons and Parmesan shavings, and season with salt and pepper. Serve immediately.