Preheat the air fryer. Set to 370°F (190°C) and preheat 3 to 5 minutes.
Dry the sauerkraut. Place the sauerkraut in a clean kitchen towel and wring out all excess liquid. Transfer to a small bowl and mix with 1 tablespoon Russian dressing.
Fill the wrappers. Lay an egg roll wrapper on a work surface with one corner facing you. Place a slice of Swiss cheese on the bottom third, top with corned beef, then the sauerkraut mixture. Divide filling evenly among 12 wrappers.
Roll and seal. Fold the bottom corner over the filling, fold in both sides, and roll tightly. Dip a fingertip in water, run it along the top corner, and press to seal. Repeat with remaining wrappers.
Air fry. Spray the basket with nonstick cooking spray. Place the egg rolls seam-side down in a single layer, not touching — cook in batches as needed. Air fry 5 minutes, flip, spray the tops, and cook 5 more minutes until golden and crisp.
Serve. Serve hot with extra Russian dressing for dipping, garnished with parsley if desired.