Make the mayonnaise by whisking together all the ingredients until fully incorporated. Transfer to a small serving bowl and serve as a side with the fried potatoes.
Heat the oil over high heat in a large, heavy pot, such as a Dutch oven, until it reaches 375 degrees F. Carefully drop half of the potatoes into the hot oil and cook until nicely golden brown, about 8 minutes. Remove potatoes with a slotted metal spoon onto a plate lined with a double layer of paper towels. Season generously with salt and sprinkle with parsley. Serve the first batch and repeat the frying process for the second batch.