The former prime minister was to have led an "in conversation" event with Mr Clinton at the Royal Institution of Great Britain in central London, but sent an apology after he had to rush to his father's bedside.
In a statement, Mr Blair said: "I am really sorry that I am not able to be with you tonight. My father has been taken seriously ill and I wanted to be with him and my family tonight.
"I am a great believer in Third Way progressive politics. I still believe it is the answer in these troubled times and can think of no-one better to explain it than my great friend, President Clinton."
Mr Blair's father Leo was brought up in Govan in Glasgow by foster parents James Blair, a shipyard worker, and his wife Mary, and took their surname. He went to Govan High School, then studied law, becoming a barrister and later a university lecturer.
Meanwhile, President Clinton revealed he had had surgery to remove a cataract from his right eye, telling the audience of Labour members he now had the "eye of a 20-year-old".