If you're using new potatoes, just scrape them to remove the skin. If not, peel the potatoes.
Boil them in salted water.
Remove the leaves from the parsley. Add the stems to the boiling water; the leaves will be used at the end.
The potatoes should not be overcooked, so boil them until slightly firm to the bite. Here, they got 12 minutes of boiling without a lid, and then the lid was put on, and the heat turned off. Finally, they stayed in the hot water for 8-10 minutes.
Heat a sauté pan over medium-high heat (about 10 out of 14 on my stove).
Add butter to the pan, and when it sizzles, add the potatoes directly from the pot, letting some of the boiling water come with them.
Chop the parsley leaves and add them to the pan. The potatoes only need to be "coated" in the butter/parsley mixture.
Season with salt and pepper and serve immediately.