Árpád Bárány took up fencing as a young boy and was able to develop his skills after World War II. His training as a physical education instructor interrupted his competitive career, but he nonetheless became the national winter pentathlon champion in 1956. He then focused on épée fencing and won his first national team title in that category in 1957. That same year he made his major international début for Hungary at the World Championships, where he was runner-up in both the individual and team events. He repeated team silver at the 1958 Worlds, and won both the individual and team titles at home that year. In 1959 he achieved team gold at both the Worlds and the nationals, but dropped to third individually in Hungary. His next stop was the 1960 Rome Olympics, where the Hungarians lost the team bronze medal match to the Soviet Union.
In the lead-up to the 1964 Tokyo Olympics, Bárány took team bronze at the Worlds in 1963 and individual bronze and silver at home in 1961 and 1964 respectively. At the Games, however, he captured gold in the team event alongside Győző Kulcsár, Zoltán Nemere, Tamás Gábor, and István Kausz. He retired from active competition in 1968 and focused full-time on coaching, first in Iran, then at clubs in Hungary, where he also served as a referee.
Hungary