Cut the new potatoes into quarters (or sixths if they are large).
Boil the potatoes in a large pot of well-salted water until they are tender but not mushy. Drain and set aside to cool. (Note: new potatoes have thin skins and cook faster than baking potatoes. Test for doneness after 10 minutes).
Meanwhile, combine the mayonnaise, mustard and cider vinegar. Mix well.
In a large mixing bowl, combine the potatoes, eggs, celery, green onion, parsley, salt and pepper.
Stir the dressing into the potato mixture until well combined. Taste and adjust the seasoning with more salt and pepper if needed.
Using a potato masher or the back of a wooden spoon, smash the potatoes slightly to get a creamer consistency.
Refrigerate to blend the flavors before serving.