Bruno Parma achieved success as a high school student, when he won the Yugoslav Youth Chess Championship for three years in a row. In 1961, he became the world junior champion in The Hague, making him the youngest international master in the world at the time.
As a member of the Yugoslav national team he competed in eight Olympics and is the holder of six Olympic medals. He was awarded the Grandmaster title by FIDE in 1963 on the basis of the Beverwijk Grand Tour. After finishing his competitive career, he worked as a coach, was the selector of the Yugoslav national team and the first selector of the Slovenian national team.