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, then immediately reduce the heat to a gentle simmer and cook until the potatoes are tender, about 10 minutes.
While the potatoes are cooking, mix the relish, mustard, parsley, celery and onion in a bowl and season with salt and pepper. Mix in the mayonnaise. Let this sit for at least 20 minutes while the potatoes are cooking so the flavors can marry.
Drain the potatoes and transfer them to a large bowl. While the potatoes are hot, add the vinegar and use a rubber spatula to toss gently to combine so the potatoes absorb all that vinegar flavor.
Toss the potatoes with the dressing base and the cheese; season with additional salt and pepper. Cover with plastic wrap and refrigerate until chilled and to let the flavors meld, about 30 minutes. Once ready to serve, garnish with more parsley.