Boil the potatoes in salted water until tender, about 15 minutes. Drain and let cool slightly.
In a small bowl, whisk olive oil, vinegar, mustard, salt, and pepper.
Place warm potatoes in a large bowl and pour the dressing over. Toss gently to coat.
Add red onion, parsley, and eggs if using. Stir lightly until everything mixes together.
Serve warm or chilled, depending on your mood.