Place the cubed potatoes into a large pot of salted water. Bring to a boil and cook until the potatoes are tender and easily pierced with a fork, about 15–20 minutes. Before draining, reserve 1/4 cup of the cooking water.

While the potatoes cook, peel the garlic cloves and remove any green sprouts from the center. Place the garlic and 1 teaspoon salt into a mortar and pestle and crush until a smooth garlic paste forms. If you don't have a mortar and pestle, a garlic press or microplane works well too.

Drain the potatoes and mash with a potato masher, food processor, or stand mixer until smooth. Gradually add garlic paste and red wine vinegar. Lastly, add the olive oil to reach a smooth texture. If the potatoes seem dry, you can also add a little of the reserved cooking water; however, I didn't need it.

Transfer to a serving bowl and drizzle with olive oil. Garnish with fresh parsley and serve at room temperature with warm pita bread, vegetables, or fish.