Place the potatoes in a pot with salted water and bring to a boil. Cook for about 15 minutes, until tender but still firm. Drain and set aside.
To make the aioli, crush the garlic with a pinch of salt in a mortar until a smooth paste forms.
Add the egg yolk and mix continuously in a circular motion.
Stir in the lemon juice, then slowly drizzle in the olive oil, mixing constantly, until the sauce thickens and becomes creamy.
Alternatively, place all aioli ingredients in a tall container and blend with an immersion blender until fully emulsified.
Transfer the potatoes to a large bowl and gently fold in the aioli, taking care not to break the potatoes.
Taste and adjust seasoning if needed.
Move to a serving dish, sprinkle with chopped parsley, and refrigerate for 1 hour before serving.