Ödön Téry competed for BTC from 1907 in track, field, and gymnastics disciplines, but he was most successful in gymnastics, primarily on the parallel bars and horizontal bar. In May 1912, he was injured and he seemed unable to attend the Stockholm Olympics. He quickly recovered, however, and became a member of the Hungarian team that finished second.
In 1914, Téry received a degree in mechanical engineering in the Budapest József Műegyetem. A few years later he emigrated to the United States. His father, Ödön Téry, Sr., was a leading figure of the Hungarian gymnastic movement.