Cook pasta according to package directions. Brush chicken with 2 tablespoons of olive oil and sprinkle with grill rub. Grill chicken, uncovered, over medium heat until juices run clear, 5-8 minutes on each side. Remove from grill to a rimmed platter. Sprinkle with lemon juice; let stand 10 minutes. Drain pasta; rinse with cold water.
Remove chicken from platter to a cutting board and cut into strips; pour juices into a large bowl. To the juices, add vinegar, mayonnaise, mustard, salt and pepper; whisk well. Gradually whisk in oil. Add pasta, celery, onion, red pepper and oregano; toss to coat. Arrange pasta over lettuce-lined plates. Top with chicken. If desired, sprinkle with parsley.