Daniel Muston, 40, Ryan Peter Marshall, 30, and Anthony Raymond Mitchell, 31, were charged after allegedly performing the salute near the museum in Darlinghurst on October 13.
On Tuesday, the three men pleaded not guilty to charges of behaving in an offensive manner in public and knowingly displaying Nazi symbols without excuse.
Daniel Muston's lawyer Bryan Wrench said there was "a bit of unfairness" in the prosecution.
Muston's lawyer, Bryan Wrench, told Sydney's Downing Centre Local Court that Premier Chris Minns had forgiven his predecessor for wearing a Nazi uniform to a party as a young man but was now pursuing an "authoritative use of power" against his client.
"We think there's a bit of unfairness in this," Mr Wrench said.
The solicitor added the men had no links to Nazi groups or history of sympathising with the fascist ideology and had no related paraphernalia on them at the time of the alleged offending.
"There were no swastikas and no Nazi uniforms - unlike the former premier, Dominic Perrottet, or Prince Harry," he said.
Mr Perrottet apologised for wearing a Nazi uniform to his 21st birthday party after  the incident was revealed at the start of the year, adding that he was "deeply ashamed" over his behaviour.
The men's arrest earlier this month came amid community tension over the escalating conflict in Gaza, including multiple examples of alleged anti-Semitic rhetoric and behaviour.
Mr Minns told reporters at the time the full extent of the law should be applied to the trio.
Outside court, Mr Wrench said the only uniform Muston was wearing at the time of the alleged incident was a hi-vis vest.
The lawyer for Ryan Marshall said the case was a "misunderstanding".
"He's not a Nazi, he's never had a swastika," he said.
"More importantly, he's never worn a uniform, owned a uniform or intended to wear a uniform.
"He's not guilty of the charges. He's not part of a protest."
Marshall's lawyer, Upol Amin, said the case was a "misunderstanding" and not linked with antagonism over the Gaza conflict.
"We're in very sensitive times at the moment and one thing that needs to be made clear is that this matter has nothing to do with what's happening with Israel and Palestine," he said outside court.
"We hope that the misunderstanding will be cleared up by the courts in due course."
The case is due to be heard next on January 23, with the three men to remain on bail.
