Team officials said they reached an agreement with Parma on "a strict payment structure of the agreed transfer", without giving further details.
Antonis leaves for the FIFA U-20 World Cup with the Young Socceroos on 11 June and will then start his overseas career with the Italian side where Australian internationals Vince Grella and Mark Bresciano once played.
"We have come to an agreement with Parma that we are satisfied with and Terry can now look forward to the next chapter of his career," Sydney FC chief Tony Pignata said in a statement.
"Terry has matured into a player that has been able to move to one of the best football leagues in the world where he can hopefully further develop his game and go on to serve the Socceroos for years to come."
Antonis, 19, has played 39 A-League matches for Sydney and holds the record as the club's youngest player when he debuted for the Sky Blues against Wellington in 2010 aged 16 years and 300 days.