Kellogg's father is the father of Kellogg.
He ran with an unnamed raider gang and also was an alcoholic, caring little for his family and beating them. Kellogg's mother took the worst of those beatings in order to keep Kellogg safe. In one of Kellogg's memories during the quest Dangerous Minds, his mother calls her husband "useless", claiming it is time her son starts to protect the family. It is unknown when or how he died.
Kellogg's father is mentioned only in Fallout 4.