Born in the village of Monteleone d'Orvieto, about 35 km southwest of Perugia, he studied with Cesare de Sanctis at the Accademia Nazionale di Santa Cecilia in Rome between 1891 and 1899.

He started work as a conductor in Italy and Paris, before moving to the United States.

In 1925 Parelli returned to Italy and became artistic director of the newly formed Unione Radiofonica Italiana (Italian Radio Union; now RAI) in Milan.

In 1926, I dispettosi amanti became the first opera to be broadcast in its entirety on Italian radio.

He died in Monteleone d'Orvieto on 26 December 1944 and lies buried in the cemetery of his native village.