Heat the olive oil in a saucepan. Cook the onion until soft, add the garlic, paprika, and cayenne, and cook for 30 seconds.
Add the crushed tomatoes and simmer for 12 minutes. Blend until smooth, stir in the vinegar, and season to taste. Mix the mayonnaise and garlic for the aioli.
Fry the potatoes:
Heat the oil to 300 F (150 C). Fry the potato cubes in batches for 6 to 7 minutes, until tender but pale. Drain on a wire rack and cool for 10 minutes.
Raise the oil to 375 F (190 C). Fry the potatoes again in batches until deep golden and crisp. Drain and season with salt.
Serve:
Arrange the hot potatoes on a serving plate. Spoon over the bravas sauce and add the garlic aioli.
Finish with parsley if desired and serve immediately.
Notes
Fry the potatoes twice for the best contrast of crisp outside and fluffy center.