The announcement comes a day after Mexico beat Panama to win the CONCACAF Nations League bronze medal after a 3-0 loss in the semi-finals on Thursday to the United States, who went on to retain their title.
FMF announced that Jaime Lozano, who led Mexico to a bronze medal at the Tokyo Olympics in 2021, will take over as interim manager.
"We thank Rodrigo and Diego for their work and dedication during this time and wish them every success in their future projects," FMF added in a statement.
Ares de Parga and Cocca had only been appointed in February in a shake-up by FMF after the country's elimination in last year's World Cup group stage for the first time since 1978.
Cocca had a contract until the 2026 World Cup but local reports said his tenure was subject to Mexico's results in this year's continental tournaments.
Mexico will face Honduras in their Group B opener in the Gold Cup, which they will be looking to win after losing the final to the U.S. in 2021.
