The council moved on Wednesday to prohibit the planned processions having considered expert advice from Police Scotland.
It followed trouble at parades held in the city over the past two weekends, with police responding to "significant disorder" at a march in Govan on August 30.
The following weekend, 11 people were arrested as two Republican marches were held through Glasgow city centre, with a police officer injured by a pyrotechnic thrown by loyalist protesters.
Partick Orange and Purple District 15 had been planning to march on Sunday.
"I believe that the right to march is an important part of our democracy but those who are abusing that right I think are putting it into jeopardy for others.
"It is also vital that the rights of the majority of law-abiding citizens are protected and given priority."
She added: "So I think Glasgow City Council has taken the right decision.
"Obviously it takes those decisions in light of the advice it receives from the police.
"I think there are longer-term questions about whether there are changes required to the law and we will continue to have that dialogue with Glasgow City Council."
A statement released by the council following its decision on Wednesday read: "The council has always been clear that the law expects it to facilitate public processions, including those that some people oppose or find offensive.
"However, the right to march has to be balanced against the rights of people and communities across Glasgow."
It added: "The city has already witnessed an unacceptable level of disruption and disorder associated with parades and counter-protests in recent weeks.
"It is clear, both from the intelligence gathered by police and the tone of comments made by supporters and protesters, that tensions are high and the situation threatens to deteriorate further.
"The council directly appeals to those who would have taken part in these marches, or who planned to mount protests against them, to comply with the orders made and not bring further disruption to city streets."