Boil the potatoes in salted water for 8-10 minutes until they’re fork-tender but still holding their shape.
Drain the potatoes and pat them dry.
Add oil to fill a heavy pot or deep fryer about an inch deep, and heat it to 350 F.
Fry the potatoes in batches for 8-10 minutes until golden and crispy.
Drain the potatoes on paper towels, and season them with salt.
For the pimentón sauce, whisk the mayonnaise, tomato paste, pimentón, vinegar, garlic, and cayenne. Season with salt.
For the alioli, whisk the garlic and egg yolk together. Slowly drizzle in the oil while whisking constantly until emulsified. Whisk in lemon juice and salt.
For the green sauce, blend the parsley, oil, garlic, vinegar, and salt until smooth.
Transfer the potatoes to a serving bowl. Drizzle them with pimentón sauce and alioli. Top with dollops of green sauce.