New South Wales police are set to do DNA and dental analysis to help identify the man.
9 News report that police believe the man has been dead for years, and are treating the case as a homicide.
On Saturday, human bones were discovered just 2km away from Leveson's burial site.
A motorist came across a large bag filled with the remains at about 4.45pm, after stopping for a toilet break at the Royal National Park in Sydney.
At the same time, Leveson's parents Mark and Faye were paying their respects at their son's burial site.
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Talking to Seven News, Leveson said, "We hope that some other family now will be able to find their loved one."
"When you hear remains are found or a body is found you just wonder or hope that this could be our turn."
Leveson's son went missing back in 2007 from a Sydney nightclub.
A decade later his former boyfriend Michael Atkins was acquited for his murder, with the agreement to lead authorities to the body.
In May 2017 Leveson was found.
If you have any information regarding the current findings, police are urging people to contact Crime Stoppers.