Mystical Meatballs

1 1/2 lb small (1 1/2- to 2-inch) boiling or fingerling potatoes
3 Tbsp unsalted butter, cut into pieces
2 Tbsp chopped fresh flat-leaf parsley
2 Tbsp + 1/2 tsp kosher salt

Peel potatoes and cover with cold water by 2 inches in a 5-quart heavy saucepan. Add 2 Tbsp salt. Simmer, uncovered, until just tender, 15 to 20 minutes. Drain in a colander and toss in a bowl with butter, parsley, and 1/2 tsp kosher salt.