Preheat grill to 350 degrees, you will cook these on a covered grill. The temperature of your grill should read 350-375. Reduce flames to low to avoid scorching.
In a large bowl, toss the potatoes with oil and then transfer to a grill basket. Place on the grill, and cover with lid. Every 5 minutes or so, check the potatoes and give them a quick shake. Replace lid. The potatoes will take anywhere between 15 to 20 minutes, depending on their size and the heat of the grill. A little bit of char is a good thing. A little bit. Test the potatoes by inserting a knife or a fork into their flesh, it should slide easily with no resistance.
Once potatoes come off the grill, all them to cool only to the point where you can handle them. Cut each potato in half or quarters, should be largish bite-sized pieces, and toss into a large bowl. Toss still warm potatoes with white wine vinegar and a generous pinch of salt. Allow them to cool a bit longer at this point.
Directly to the potatoes, add mayo, mustard, and red onions. Toss well to combine. Taste for seasoning, add salt and pepper, as needed, and fresh chopped parsley.