Cook the potatoes either by steaming in a steamer basket until just barely fork-tender (about 15 minutes) or pressure cooking on high for 4 minutes (use 1 cup water in bottom and steamer basket; turn off heat after pressure cooking for 4 minutes, let sit for 5 minutes before releasing valve). I separated the blue potatoes while cooking so they wouldn't stain the others. Let cool.
Add potatoes to bowl with celery, green onions, and parsley.
Add all dressing ingredients to a jar with a tightly fitting lid and shake until well blended. Pour over salad and gently toss.
Keep refrigerated until ready to serve. Note that vegan and egg-free mayo is much safer for outdoor picnics and BBQs.