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Her parents are Michael and Carole Middleton.
Catherine, Duchess of Cambridge's parents are known as Carole and Michael Middleton, the owners of the Middletons' family business, Party Pieces, which sells party supplies and kits for childrens' parties for all of the United Kingdom.
Catherine Middleton's or Duchess Kate's parents have no special name except their usually-known names of Carole and Michael Middleton.
Her mother in law is the late Diana, Princess of Wales who was mother to her husband, the Duke of Cambridge (Prince William.) Camilla was never a "mother in law" to the Duchesss of Cambridge as she is not Prince William's mother.
Carole Middleton's birth name is Carole Elizabeth Goldsmith.
Carole Middleton
Michael Middleton and Carole Middleton
Carole and Michael Middleton.
Kate Middleton's parents are Carole Elizabeth Middleton and Michael Francis Middleton.
Carole Troll goes by The Duchess of Troll, and Coco.
Carole and Michael Middleton.
Carole Elizabeth Middleton (nee Goldsmith) is Kate Middleton's mothers name.