Méndez Padrón was born in Matanzas, Cuba, where his father, José Antonio Méndez Valencia, was director of the Matanzas Chamber Choir, and his mother, Lilian Padrón, was a dancer and choreographer. Young José, like so many other Cubans, at first wanted to become a baseball player, but his father also conducted a local children's choir in which José sang. Participating in a children's vocal festival, the Festival Infantil Cantándole al Sol, he settled on a musical career and entered the Escuela Profesional de Arte de Matanzas. While he was there, his teacher, Enrique Pérez Mesa, was designated director of Cuba's Orquesta Sinfónica Nacional and appointed Méndez Padrón director of the school's chamber orchestra. At the time, Méndez Padrón had no experience as a conductor, but he applied his experience of watching his father conduct his children's choir, and succeeded. He entered the Instituto Superior de Arte, majoring in orchestral conducting. He studied choral conducting with María Felicia Pérez and orchestral conducting with Jorge López Marín. Méndez Padrón went on for further studies at the Mozarteum University of Salzburg, with Jorge Rotter and Peter Gülke, and at the Carnegie Mellon School of Music at the University of Pittsburgh and the Jacobs School of Music at the University of Indiana, the latter with Ronald Zollman.
In 2007, while Méndez Padrón was a student at the Instituto Superior de Arte, a statue of Mozart was unveiled in Havana. Méndez Padrón organized a group of his fellow students from that school and from the Escuelas Nacionales de Arts to perform at the statue's unveiling, and in the audience was Salzburg Mozarteum Foundation president Johannes Honsig-Erlenburg. Honsig-Erlenburg offered help in turning Méndez Padrón's new orchestra into an ongoing operation, and the Havana Lyceum Orchestra was born. Méndez Padrón, as conductor, was paid $55 a month, but otherwise, the musicians were all volunteers. Méndez Padrón also founded the Orquesta de la Universidad de los Artes, where he was on the faculty, and served as deputy director of the Orquesta Sinfónica Nacional de Cuba. In 2016, Germany's Balthasar Neumann Ensemble and its patrons assisted the Havana Lyceum Orchestra in turning professional. In the late 2010s, the orchestra, with Méndez Padrón on the podium, rose to international prominence, recording the album Mozart in Havana (2017) with pianist Simone Dinnerstein and then touring several festivals in Germany. Méndez Padrón and the orchestra released the album Mozart y Mambo: Cuban Dances in 2022 on the Alpha label. ~ James Manheim, Rovi