The correct way to write the sentence so that it uses parallel structure is
A. Flappers were women in the 1920s who bobbed their hair, cut their skirts shorter, and danced the Charleston.
This parallel structure requires the subject of the structure to be the same.
In this case the ones who bobbed their hair, cut their skirts shorter, and danced the Charleston, were always the women. The structure in joined by the coordination conjunction and.
In B;C; and D two of the subjects are the women and one is the skirts. The structure of the parallelism is broken.