In a bowl, mix together the cornichons or relish, mustard, honey, celery, lemon zest and juice and red onion. Season with salt and pepper and let sit while the potatoes cook.
Put the potatoes in a large saucepan and add enough cold water to cover by 1 inch. Season the water generously with salt. Bring to a boil over medium-high heat. Reduce the heat to a simmer and cook until the potatoes are tender, about 10 minutes.
Drain the potatoes in a colander, put back in the saucepan, add the vinegar and, using a rubber spatula, toss gently to combine.
Add the mayonnaise, chives and parsley to the celery-onion mixture and stir to combine. Add the warm potatoes to the dressing and toss. Season to taste. Garnish with extra chives and parsley.