Leave the skins on the potatoes and cut them in halves or quarters, depending on how big the potato is - you'll want pieces that are roughly the same size, about one inch. Rinse them thoroughly under cold water.
Place the potatoes in a medium to large pan, then add enough cold water to cover them. Season generously with salt, and place over medium heat. Bring to a gentle boil.
Cook until the potatoes are tender all the way through when gently pierced with a fork.
While the potatoes are cooking, add the olive oil, minced parsley, chopped capers, mustard, and vinegar to a bowl and whisk to combine. Season with salt and pepper, and set aside.
Add the sliced onions to a large bowl and set aside.
When the potatoes are cooked, remove the pan from the heat and drain the hot water.
Pour the drained hot potatoes directly into the bowl with the onions and let it sit for about a minute.
Then, pour the dressing over the potatoes and stir gently to coat.
Let the potato salad cool to room temperature and stir again a few times before covering and refrigerating.
Serve at room temperature or cold!