For the past 11 weeks, loyalists have walked through city centre to protest against changes to the council's policy on flying the Union flag.
It has led to criticism over how the demonstrations are handled by the authorities and calls for the commission to intervene.
Mr Osborne said they have dealt with 30 or so parades which have been notified, but explained that the Saturday afternoon demonstrations have not.
"First of all the protests that have been related to flags don't come under the Parades Commission, the Parades Commission has responsibility for parades and protests related to parades so the bulk of issues about flags have not been for the Parades Commission," he said.
"If a parade is not notified the legal advice we have is very straight, if it's not notified it is an unlawful procession and it is then a policing matter to manage under public order."
It [Saturday march] has not been notified to the commission, and if it is a procession as the police say and is an unlawful and illegal procession it needs to be considered by the police.
Chief Constable Matt Baggott said the laws surrounding unnotified parades may need to be reviewed to provide more of a deterrent against illegal demonstrations.
"I remain concerned that the current Public Processions Act does not provide sufficient deterrence to those breaking the law," said the PSNI chief.
"The required standard of proof to convict is high and the Act would benefit from review."
Sinn Féin MLA Gerry Kelly responded by saying: "Any un-notified parade or protest march is illegal and therefore anyone participating in them is breaking the law.
"The PSNI need to act on this fact and stop facilitating law-breaking."
Local MLA Conall McDevitt said the flags protestors have "long since forfeited their right to peaceful protest".
"The illegal flag protests are now a policing matter," the SDLP politician continued. "The protestors have long since forfeited their right to peaceful protest.
"What they are engaged in is illegal and unjustified."
The first Orange Order parade of the marching season took place peacefully over the weekend - however there are concerns that this year could be volatile.
A senior member of the organisation revealed they may sideline the Parades Commission this summer by not notifying it about parades.
Mr Osborne said he does not think that will happen.
He continued: "I would be really surprised if that was a policy direction the Orange Order wanted to take, first of all because where they should be at is upholding the law.
"I think they and everybody else in civic society should be trying to reduce tensions around parading and finding agreement and it's clear an unlawful parade would raise tensions."
A disrupted marching season is not inevitable, nor is the violence which we have seen on our streets over recent months.
Stewart Dickson, Alliance MLA, said parading groups cannot be allowed to flout the law.
"The law is clear," he said. "Those who organise or participate in a parade that has not been notified through the appropriate channels are committing an offence.
"We cannot allow the law to be flouted in Northern Ireland. It must be made clear to those who propose to ignore the notification requirements that they face prosecution.
"Inflammatory statements made in recent weeks have not helped matters."
Mr Osborne added that people have a choice this summer to avoid scenes of violence on the streets of Northern Ireland, claiming fears of a turbulent marching season are not inevitable and calling for increased dialogue to diffuse tensions following months of flag protests.
"People have a choice. Recent events have demonstrated what happens when poor choices are made - there is disruption, disorder and rioting on our streets," he said.
"This summer we want people to make different choices, a choice of engagement, a choice of respect for others, a choice of law and order. Everyone has these choices in front of them, parading organisations, local communities, politicians and wider civic society."
The marching season comes to a climax on 12 July with thousands of loyal order members taking part in parades across NI to mark the Battle of the Boyne.
It celebrates the victory of William III over the Catholic James II in July 1690.
Mr Baggott added: "I hope that parades will take place safely and with respect for all cultures and backgrounds."