A High Court judge in Belfast had ruled that the PSNI should not have facilitated illegal parades, and was wrong to think it did not have legal powers to stop them.
The ruling was quashed after an appeal.
The UK Supreme Court has now been asked to overturn the decision.
There were widespread loyalist protests across Northern Ireland after Belfast City Council decided not to fly the Union flag at the City Hall every day of the year in December 2012.
The protests included a weekly parade between east Belfast and City Hall, which passed the nationalist Short Strand area.
Critics accused the police of being too soft as some of the protests descended into violence.
The PSNI insisted it did not have the legal powers to stop them, and warned that doing so might have made the situation worse.
In April 2014, a judge in the High Court in Belfast upheld a claim by a Short Strand resident, who cannot be named to protect his safety, that the PSNI had failed in its legal duty to stop the parades.
The judge said the police had not understood the full scope of their powers and had facilitated illegal and sometimes violent parades, in breach of their legal duties.
That ruling was appealed by then Chief Constable Matt Baggott, and quashed by the Court of Appeal.
The legal battle moved to the UK Supreme Court in London on Tuesday.
A lawyer representing the unnamed resident, told five judges the PSNI had failed in its duty to enforce the law as set out in the Public Processions Northern Ireland Act (1998).
She argued that the Court of Appeal had no grounds to interfere and overturn the original trial decision as "the evidence before the court was unimpeachable".
The lawyer said there were "evident flaws" in the Court of Appeal decision and its analysis of the evidence.
Lord Neuberger, President of the Supreme Court, asked her: "So you say that we should go back to the trial judge and, provided we concluded that there was evidence on which he could fairly reach the decision that he did, that we should reinstate his decision?"
She replied: "Yes."
But a lawyer representing the PSNI said the Court of Appeal had made the correct decision.
He said the context of the time the protests had taken place was important.
He told the court the police had been dealing with a series of sporadic unplanned protests at more than 80 locations across Northern Ireland.
He insisted there had been "no error of law" and that the police fully understood that they could have stopped the parades, but took an operational decision not to so.
He said the police believed that "robust action" could have made the situation worse and took a deliberate decision to secure evidence of offences committed and to prosecute offenders afterwards.
The lawyer referred to research by Queen's University, which recorded that 306 people had been convicted of flag-protest-related offences across Northern Ireland in the months after the protests.
"Many would have considered the police approach to policing these parades was successful, and that their policy of de-escalation succeeded," he said.
The judges said they would announce their decision "in due course".
A resident of the Short Strand area of east Belfast has challenged a decision to overturn a court ruling that the PSNI was wrong not to stop union flag protests.