For the sherry-saffron aioli: Combine the sherry vinegar and saffron threads in a small bowl and let sit for 5 minutes to rehydrate the saffron.
Combine the mayonnaise, shallots, anchovy paste, garlic, 1/2 teaspoon salt and the saffron-vinegar mixture in a food processor; process until smooth. Transfer to an airtight container and refrigerate until ready to use, up to 2 days.
For the spicy tomato sauce: Heat the oil in a medium saucepan over medium-high heat. Add the shallots and garlic and saute until softened, 3 to 4 minutes. Stir in the tomato paste and 1 teaspoon salt and cook for 2 minutes, stirring occasionally. Add the chopped tomatoes and red pepper flakes. Bring to a boil, then reduce the heat and simmer until the sauce thickens, 15 minutes. Serve immediately, or cool and refrigerate until ready to use, up to 2 days.
For the potatoes: Place two baking sheets on oven racks. Preheat the oven to 425 degrees F.
Cut the potatoes into halves or quarters, depending on size, to get roughly 1-inch pieces. Toss the potatoes with the olive oil and 1 tablespoon salt in a large bowl. Carefully pour half of the potatoes onto the first baking sheet; they should sizzle as they hit the pan. Repeat with the remaining potatoes on the second baking sheet. Roast, stirring halfway through, until the potatoes are tender and crispy, 40 to 45 minutes. (If some of the potatoes do not move when you stir them, do not force them; they will release from the pan when they have crisped up.) Let cool slightly.
Spread one third of the tomato sauce on the bottom of a serving dish. Top with half of the potatoes. Spoon another third of the tomato sauce over the potatoes, and then top with the remaining potatoes and the rest of the tomato sauce. Dollop with the aioli, and sprinkle with the parsley and some paprika. Serve any extra aioli on the side.