Mr. Kavinsky is the father of Joseph Kavinsky. He does not appear in the books.
In The Dream Thieves, when Ronan wakes up in Kavinsky's home cinema after crashing the Pig, Ronan asks Kavinsky if his father is a dreamer. Joseph deflects this with the question turned back on him. This leaves it up to speculation whether or not Mr. Kavinsky is a dreamer.
Ronan mentions in The Dream Thieves that he has heard rumors that Joseph tried to kill his father and was sent to Aglionby because of it. Joseph does not outright confirm or deny this, saying that he "only does what he means to." He goes on to say that his father was the one who tried to kill him instead of the other way around and "he doesn't always do what he means to." All these contradicting and muddled points leave nearly everything about Mr. Kavinsky up to speculation and theory.