Place potatoes in a large saucepan; cover with water. Stir in Kansas City Dry Rub and 1 teaspoon salt. Bring to a boil. Reduce heat; simmer 15 minutes or until tender. Drain.
Combine Kansas City Barbecue Sauce, oil, vinegar, and mustard in a large bowl; stir with a whisk. Add warm potatoes, and toss to coat. Add remaining 1/2 teaspoon salt, celery, onion, parsley, and pepper; toss to combine. Let stand 20 minutes; serve at room temperature.