Put the potato in a medium saucepan, add enough water to cover and salt generously. Bring to a simmer, then cook until fork-tender, about 30 minutes. Add the garlic, cook for 2 minutes, then drain. When the potato is cool enough to handle, remove and discard the skin and cut the potato into chunks.
Transfer the cooked potato and garlic to a food processor. Add the chickpeas, oil, 1/2 cup cold water, vinegar and 1 teaspoon salt. Process, scraping down the bowl a few times, until the mixture becomes a creamy puree. Transfer the dip (serve at room temperature or chilled) to a serving vessel, drizzle oil over the top and sprinkle with pepper flakes and parsley. (The dip can be made a day ahead and refrigerated.) Serve with cucumber rounds and pita chips for dipping.
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