Put the potatoes in a medium pot and add cold water to cover; bring to a boil. Lower the heat to medium and cook until barely fork-tender, 12 minutes. Drain.
Meanwhile, put the parsley, mint, pistachios, oil, garlic and 1/4 teaspoon salt in a food processor and process until finely minced (see Cook's Note). Transfer to a large bowl and stir in the Parmesan until blended.
Toss the pesto with the warm or room-temperature potatoes and serve.
Very lightly pack herbs into measuring cups (otherwise the proportions will be slightly off). Do not over-process the pesto; if you do, the color will be dull green instead of bright.