Harrell was born to musician parents in New York City: his father was the baritone Mack Harrell and his mother, Marjorie McAlister Fulton (1909–1962), was a violinist. At the age of nine, he began cello studies. When he was 12, his family moved to Dallas, Texas, where he studied with Lev Aronson (1912–1988). After attending Denton High School, Harrell studied at the Juilliard School in New York with Leonard Rose and then at the Curtis Institute of Music in Philadelphia with Orlando Cole. In 1961, when he was 17, he made his debut at Carnegie Hall with the New York Philharmonic Orchestra as part of a Young People's Concert.
In 1960, when Harrell was 15, his father died of cancer. In November 1962, when he was 18, his mother died from injuries sustained from a two-vehicle crash while traveling from Denton to Fort Worth with pianist Jean Mainous to perform a recital; she was violinist in residence (faculty) at the University of North Texas College of Music.
Just before his mother died, in April 1962, Harrell had withdrawn from Denton High School in his junior year to advance to the semifinals of the Second International Tchaikovsky Competition in Moscow.
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