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Heat a tablespoon of oil in a small saute pan over medium heat. Add the onion and garlic and cook until soft, about 5 minutes. Add the paprika and cook for 30 seconds. Let cool slightly.
Combine the onion mixture, mayonnaise, tomato, hot pepper sauce, and vinegar in a food processor and process until smooth, season with salt and pepper. Scrape the mixture into a bowl, cover and refrigerate for at least 30 minutes to allow the flavors to meld.
Heat 2-inches pure olive oil or canola oil in a large high-sided, heavy-bottomed skillet until it begins to shimmer. Add the potatoes, season with salt and pepper and cook until golden brown on all sides. Remove the potatoes to a paper towel-lined plate for a moment to drain the excess oil. Carefully remove the hot baking sheet from the oven, put the potatoes on the pan in an even layer and bake until crisp, about 10 minutes. Remove the pan and season the potatoes with a bit more salt. Transfer to a platter and drizzle with some of the aioli and garnish with parsley leaves. Serve hot.