Career
Parra was successful as an amateur in Colombia, winning the Novatos classification for new riders or riders riding their first edition of the race, and finishing 14th in the 1979 Vuelta a Colombia and then the General classification in the 1981 Vuelta a Colombia. He also competed in the individual road race event at the 1984 Summer Olympics. Parra turned professional for the first Colombian cycling team, Café de Colombia, in 1985.
His success occurred at the same time as his compatriot Luis Herrera.
Fabio Parra has two younger brothers who also became professionals, Humberto Parra Pinto and Iván Parra.