In a large pot, bring 6 qt. water and 2 Tbs. salt to a boil. Add the orecchiette and cook until al dente, 10 to 12 minutes. Drain the pasta and toss it immediately with 1 Tbs. of the olive oil. Let the pasta cool completely in the refrigerator. Peel and seed the roasted peppers and cut them into 1-inch pieces.
Heat 1 Tbs. of the olive oil in a small skillet over medium heat. Add the bread and fry, turning occasionally, until golden on both sides. Transfer the bread to a food processor. In the same skillet over medium heat, fry the almonds, stirring until golden, about 2 minutes. Add the almonds, tomatoes, garlic, paprika, and red pepper flakes to the processor and pulse several times. In a small bowl, combine the vinegar with the remaining 4 Tbs. olive oil. With the processor running, gradually pour in the olive oil mixture until well combined and smooth. Season with salt and pepper. Let sit for 1 hour before using.
In a large bowl, toss the orecchiette with the peppers, sauce, and the chopped parsley. Season to taste with salt and pepper. (At this point, the pasta can be held for a few hours in the refrigerator, if you like.) Put the pasta in a serving bowl and garnish with the parsley leaves (and grilled shrimp or sliced grilled meat, if using).