The stage version borrows the 2007 movie's songs by Glen Hansard and Markéta Irglová, who also played the would-be lovers in the film. The Broadway production, which premiered in 2012, won eight Tony Awards including Best Musical.
Melbourne performances begin Sept. 26 at the Princess Theatre.
Once has a book by Irish playwright Enda Walsh (Penelope), direction by John Tiffany (Black Watch), movement by Steven Hoggett (American Idiot, Black Watch) and set and costume design by Tony Award winner Bob Crowley (The Coast of Utopia, Mary Poppins).
In addition to Parsons (Jesus Christ Superstar) and Jones (Equus), the cast of actor-musicians will include Anton Berezin, Andrew Broadbent, Ben Brown, Gerard Carroll, Colin Dean, Brent Hill, Stefanie Jones, Keegan Joyce, Amy Lehpamer, Lachlan Neate, Jane Patterson, Greg Stone, Susan-ann Walker and Paul Watson.