Preheat the grill.
Mix the mayonnaise, parsley, vinegar, garlic, and mustard is a small bowl and refrigerate.
Cut the potatoes and onions into 1/2 inch slices and drizzle with olive oil.
Put the potatoes on the grill over direct medium heat. After 2-3 minutes, add the onions.
After about 6 minutes, turn the potatoes and onions. (Move the potatoes at the front of the grill to the back and vice versa, so that they'll all cook somehwat more evenly.)
After about 6 more minutes, remove the onions from the grill. Let the potatoes cook for 2-3 minutes more, then remove them from the grill.
Roughly chop the potatoes and onions and put in a large bowl.
Add the dressing and celery and toss well to coat the potatoes.
Season with salt and pepper to taste.
Best served warm, but it works at room temperature as well.