What mental illness/psyc disorder does Jeanette's father have?

Like mother, there never really is a specific illness (besides alcoholism) that father is attached to. Check out this description from GradeSaver,

"Jeannette describes him as smelling like cigarettes, whisky, and hair tonic. Rex invents credentials in order to get jobs, which he never keeps for long. A dreamer, Mr. Walls creates many fantastical stories to explain realities of life to his children. For instance, when they have to move because of bill collectors, Rex instead tells his children that they are being chased by FBI agents. Rex is plagued by drunkenness and gambling addictions."

There is a sense of paranoia and mania to him. One can speculate that his grandiose plans and fantastic stories about being chased by FBI agents could be associated with a Schizophrenia- type condition. Like mother he is also prone to mood swings which suggest a bi-polar issue.