The Irish rocker and the late country music star sing together on Torn, which is described as a song about broken family ties.
The country music track, which appears on U2's 16th album, Carnaval De Luz, was co-written by the band, Parton and Snow Patrol star Johnny McDaid.
Parton died in August at the age of 80 following a "brief battle with cancer".
Carnaval De Luz is the band's first album of new material since Songs Of Surrender (2017). Its first single, Street Of Dreams, was released in July.
The record is said to be a collection of 12 songs exploring themes of spirituality, friendship, freedom and endurance "in the face of dark, encroaching forces".
Speaking about the album, Bono said: "Roll up, roll up, roll up! Bring your purple socks, your broken hearts, your rose-tinted naivety… oh be amazed in the hall of mirrors.
"There will be loud guitars, deep bass, some ecstatic rhythms and, yes, some silver tears will make an appearance!
"Because in the near distance just outside our carnival of light there are all kinds of creatures… the same imagination that brings us our creativity can also bring us monsters."
The Edge said: "It's a messy, provocative, sacred collection of songs - and features the first-ever use of flat-wound strings on a U2 recording. Just saying."
Carnaval De Luz will be released on November 13.
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