Fill a large pot halfway with water and add 1 tablespoon salt; bring the water to a rolling boil.
While the water heats, clean the baby potatoes by scrubbing and rinsing them under cold running water.
Add the baby potatoes to the boiling water. Return to a boil and cook until the potatoes are fork-tender, about 15 minutes.
Drain the potatoes in a colander and rinse briefly under cold water to stop the cooking; drain well.
Melt 3 tablespoons unsalted butter in a small saucepan or microwave-safe bowl.
Slice the drained potatoes into wedges and return them to the pot or place in a mixing bowl.
Pour the melted butter over the potato wedges and add 1/2 teaspoon salt, 3 dashes ground black pepper, and 2 tablespoons chopped Italian parsley.
Gently toss or stir to coat the potatoes evenly, then serve immediately.