Which playable character has a campaign parallel to Arkham City's main story? Catwoman does.
Catwoman’s campaign follows Selina Kyle as she carries out a heist in Arkham City. Her missions are interwoven with Batman’s story, intersecting the main plot at several points rather than existing as a completely separate adventure.
She plays differently from Batman. Her combat emphasizes speed and agility, and she uses clawed gauntlets, bolas, and a whip. Certain Riddler trophies and challenges are also designed specifically for Catwoman and cannot be completed by Batman.
A common mix-up is confusing Catwoman with Robin or Nightwing. Both are playable in Arkham City’s downloadable challenge content, but Catwoman is the character with a parallel story campaign in the main game. The campaign was initially distributed as console DLC and later included in the PC and Game of the Year editions.