Officials organising the annual marches issued a determination which said all bands should adhere to a single drumbeat while passing the St Patrick's RC Church on Donegall Street.
However, the Glasgow Orange Defenders Flute Band is thought to have contravened the order and admitted to playing hymns while passing the venue.
During the morning march members said they performed the Land of Hope and Glory, before playing Abide with Me on the return parade at around 7pm.
Some worshippers were said to be at the church during the return march.
Following the incident, Stuart Leckenby, band sergeant with the Yoker-based outfit, issued a brief statement on Facebook.
He said: "If playing a hymn passed [sic] a Christian place of worship is wrong then we must be guilty. If asked if I would change or play anything different? Never. We are what we are."
A spokesman for the Police Service of Northern Ireland said footage of the parade would be reviewed.
He added: "If it is deemed that there has been a breach, then we will take action."
Police said most parades in Northern Ireland passed without incident, although there were a handful of arrests recorded.