Palmieri’s parents migrated to New York from Ponce, Puerto Rico in 1926 and settled down in the South Bronx where Palmieri was born. As a child, Palmieri taught himself to play the piano by ear. He attended the public school system. At age 7, his father enrolled him at The Juilliard School, where he took piano lessons. By the time Palmieri was 14 years old, he and his 5-year-old brother, Eddie, participated in many talent contests, often winning prizes. It was at this time that his godfather introduced him to the music of the Latin bands – an experience which inspired him to become a musician.
In 1943, when still only 16 years old and still in high school, he made his professional debut as a piano player for the Osario Selasie Band. He graduated from high school in 1946, and immediately went to play for various bands. He made his recording debut with the song “Se Va La Rumba” as a member of the Rafael Muñiz Band.