The Archdiocese claimed the incident happened after Canon Tom White was meeting and greeting parishioners on Saturday evening after Mass at St Alphonsus Church.
It said it all started right as the parade came by.
"Police - who had been guarding the church - were called away to deal with another nearby incident leaving the priest and parishioners defenceless," the Archdiocese posted in a statement on Facebook.
"Canon White and parishioners were subjected to vile abuse ... 'Fenian b*****d' being the most typical.
"The priest was spat upon. Spittle landed on the back of his head. He wiped it away. Another mouthful of thick spittle was spat into his eye socket. Again he wiped it away leaving his hand full of the vile liquid.
"He was then further insulted and lunged at by a man carrying a pole before police arrived to restore some kind of order."
The Archdiocese questioned the integrity of the Orange Order marches.
"Why is the Orange Order still allowed to schedule its intimidating parades on streets containing Catholic Churches at times when people are trying to get in and out for Mass?," it asked.
The Church also said: "What kind of society is it that allows ministers of religion and church-goers to be intimidated and attacked by a group which has a long history of fomenting fear and anxiety on city streets?"
A spokesman for the Grand Orange Lodge of Scotland said it said it has been made aware of the incident and added that "no members of the parade were involved in this or any of the reported incidents".
The statement continued: "The Orange Order is founded on the principle of religious liberty and respect for people of all faiths.
"We totally condemn the bigoted actions of those involved and hope that they are dealt with to the full extent of the law. We would welcome the opportunity to work with Police Scotland to identify those responsible."
Police Scotland has launched an investigation into the incident and said: "Whilst the parade was passing the church at the time, any involvement, if at all, by someone from the Orange walk, is still to be established."
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