Place the potatoes in a large pot and cover with cold water by 1 inch. Salt the water generously and bring to a low boil, partially covered, over medium-high heat. Cook until the potatoes are fork tender, 20 to 25 minutes.
Meanwhile, in a small bowl, combine the parsley and garlic.
Strain the cooked potatoes and transfer them to a towel-lined sheet tray.
In a large sauté pan, heat the olive oil over medium-high heat. Add the potatoes and cook, shaking the pan occasionally, until the potatoes are light golden brown, 5 minutes. Add 2 tablespoons of the parsley mixture, along with a pinch of flaky sea salt, and continue to cook, shaking the pan and stirring with a wooden spoon, until the garlic and parsley are fragrant and crispy, 1 to 2 minutes. Add the butter and the remaining parsley mixture.
Using a Microplane, zest the lemon directly over the potatoes. Stir to combine and cook 1 minute more. Remove from the heat and season to taste with flaky sea salt. Transfer to a platter and serve.