Danny Chapman is the older brother of Piper and Cal
Cal mentions him briefly as being a doctor and thus making his parents proud ("Lesbian Request Denied"). A family picture sits in the Chapman living room, and the extra person in the photo is presumably Danny ("I Wasn't Ready"). He was also briefly mentioned by Piper whilst in visitation with Cal and Carol Chapman, he was in the list of names she guessed to find out that her grandmother was dying. ("Low Self Esteem City")
In "Mother's Day" Piper informs Alex that her mother Carol will not be visiting her, instead opting to spend Mother's Day with her "unimprisoned children," suggesting both Cal and Danny.