Place potatoes and 2 tablespoons salt in a large pot and cover with water; bring to a boil. Reduce heat to medium-low and simmer until fork-tender, 6 to 8 minutes. Drain. Transfer to a large bowl and set aside to cool.
Set oven rack about 6 inches from the heat source and preheat the oven's broiler. Line a baking sheet with aluminum foil.
Place chile peppers cut-side down on baking sheet and lightly brush skin with oil.
Broil in the preheated oven until skin is heavily blistered and turning black, 12 to 15 minutes. Transfer to a bowl and cover tightly with plastic wrap. Allow peppers to steam as they cool, about 15 minutes. Peel and chop the chile peppers.
Whisk mayonnaise, parsley, peperoncini peppers, mustard, lemon juice, pickle relish, and remaining 1 teaspoon salt in a small bowl to make dressing.
Pour dressing over cooled potatoes. Add chile peppers, celery, and onion; mix to combine. Season salad with black pepper. Chill in the refrigerator before serving, about 30 minutes.