He began his musical career in the 1960s as a drummer and singer in several rock groups. In 1971 he combined the solfeggio and harmony studies at the Municipal Conservatory of Barcelona with his work in the band Mi Generacion, releasing an album with his own songs. Around 1976, he took an active part in Barcelona jazz music scene, which revolved around Zeleste concert hall, playing with the bands Blay Tritono and Rondalla de la Costa. In 1979, he collaborated with Maria del Mar Bonet in the Mallorcan traditional tunes album Saba de Terrer and later on, he joined the Valencian band Al Tall. At the end of the 1970's and beginning of the 1980's he played with salsa bands, such as La Sardineta and La Plateria, as well as with singers such as Ovidi Montllor, Gato Perez, Marina Rossell and Jaume Sisa.
In 1983, he moved to Madrid and began to investigate the Castilian traditional music. He created the band Mosaico recording two albums with them. The first one of them is a tribute to the Castilian folklorist Agapito Marazuela and the second one is an album with his own compositions inspired by traditional music.
From 1990, he started his solo recording career, based on the review of the Spanish folklore interpreted with modern music parameters, with influences of traditional music from other countries, Caribbean music, jazz and rock. It had an special repercussion his album Tribus Hispanas obtaining the first prize in the Villa de Madrid contest. Parra offered his own compositions inspired by the coplas and dances of the Iberian Peninsula central area. Following the path opened by this innovative album, he recorded “Viva quien sabe querer“, Radio Ecca Prize for best album of the year, 2002.
Besides a singer, Eliseo Parra has carried out ethnological field work in collaboration with Jose Manuel Fraile Gil. The result of this collaboration are the books Traditional Romancero of the province of Madrid, Madrid Oral tradition Tales, Children's Poetry in Madrid tradition, The Mayo and its Fiestas in Madrid. He has toured half the world performing with his magnificent and loyal band and has collaborated with a large part of “traditional reconstruction“ bands in the Peninsula.
Attention should also be drawn to his Suite Romance, rehearsed by the National Ballet and the Madrid Symphonic Orchestra and his work as a musician and actor for the National Company of Classical Theatre in the play Peribañez and the Commander of Ocaña by Lope de Vega.
Finally, we need to highlight his educational work over the past six years as a percussion and traditional singing teacher.