In a medium bowl, combine the mayonnaise, onion, vinegar, mustard, pickles, pepper, and giardiniera together. Stir well to combine. Cover and refrigerate. This step can be made 1-2 days ahead. 

Place the potatoes in a large pot of cold water. Bring to a boil over medium-high heat, and add approximately 2 tsp kosher salt (the water should taste salty). Cook potatoes over medium heat until just fork tender, which takes about 10 minutes. 

Drain the potatoes, and allow them to cool in the pot or in a separate large bowl until they are just warm enough to touch. 

Add the dressing, and gently toss the potatoes in the dressing until well combined. Some of the potatoes may fall apart; this is fine, but you don't want all of them to turn to complete mush, so take care during this step. 

Cover the bowl and set aside in the refrigerator until ready to serve. I recommend making the dressing 1 day ahead, and then assembling the potato salad in the morning the day you are going to serve it. Garnish with fresh parsley, if desired.