Exactly who is related to whom in the Marsh family is never consistent. Early episodes imply that Jimbo and Marvin (Stan's grandfather) are on Sharon's side of the family, whereas more recent ones imply they are on Randy's side. Being on Sharon's side makes sense for Jimbo, as he has a different last name. However, Marvin's last name is, indeed, confirmed to be Marsh. Matt Stone revealed in an interview that Jimbo Kerns is Randy's half-brother.
In early seasons, whenever the children enter the cafeteria and Chef greets them with the "Hello there children!" line, notice his name on his apron is spelled in lowercase 'chef', but the next scene when he is speaking with the children, and the focus is on him behind the servery, notice his name is spelled in uppercase 'CHEF'.