Place a large rimmed baking sheet in the oven; preheat the oven to 425 degrees F.
Wash the potatoes well, then pat them dry. Cut each potato lengthwise into 1-inch-thick wedges. Toss the wedges with 1 tablespoon of the oil, 1 teaspoon salt and some pepper.
Carefully remove the hot baking sheet from the oven and brush it lightly with oil. Arrange the wedges, cut-side-down, on the sheet in a single layer. Bake until the underside is golden brown and not sticking to the baking sheet, about 25 minutes. Use a metal spatula to loosen and flip the potatoes, then continue to bake until golden brown and crisp all over, about 20 minutes more.
Meanwhile, heat the remaining 1 tablespoon oil in a small skillet over low heat. Add the garlic and cook, stirring often, until golden, about 3 minutes. Add the paprika and cook 30 seconds, stirring. Remove from the heat and cool slightly; then stir the mixture into the mayonnaise.
Sprinkle the potato wedges with parsley and serve with the aioli for dipping.