Parke was born in Boston, Massachusetts, the son of Sarah (née Klayman), who was born in Russia, and Charles Einstein, a pawnbroker from Austria and cousin of Albert Einstein. He married actress Thelma Leeds on February 7, 1937.
He started out a newspaper reporter in his native Boston, coming into comedy via Eddie Cantor's radio show. His own show soon followed, along with roles in films. He met his wife, Thelma, while making New Faces of 1937. Parke had made good real estate investments and for years was not dependent on performing for his income. His ill health had made him limit his comedic appearances to Friars' Club roasts.
Death on stage
Parke died from a heart attack at a Friars Club of Beverly Hills Roast of Lucille Ball and Desi Arnaz on November 24, 1958 in Los Angeles, California. He had just finished his testimonal to a house full of laughter and Art Linkletter's remark, "How come anyone as funny as this isn't on the air?", when Parke slumped onto Milton Berle's lap at the event. Berle asked "Is there a doctor in the house?"; this remark was met with laughter, as the crowd was unaware that Berle was being serious. Emcee Art Linkletter then directed crooner Tony Martin to sing a song to divert the crowd's attention; Martin's unfortunate choice was "There's No Tomorrow." According to the Los Angeles Times account, Desi Arnaz said: "This offering meant so much to me. Now it means nothing. Please, everyone, pray to your God that he will be saved."
He was carried backstage where five specialists who were Friars Club members worked feverishly to save his life. One doctor used his pen knife to make an incision in Parke's chest for open heart massage. Another used the ends of an electric cord as a makeshift defibrillator to shock his heart back to life; only the left side responded with a feeble rhythm. The doctors continued the open heart massage while waiting for the rescue squad. Despite two hours of intense medical effort, Parke was pronounced dead at 1:20 AM local time. He was 54 years old.
His children include the comedians Albert Brooks and Bob Einstein (a.k.a. Super Dave Osborne), and advertising creative/actor Clifford Einstein. By his first marriage to Lillian Anshen, he was the father of Charles Einstein, a writer.