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16 July 2015 Last updated at 07:09 BST
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The group wants to hold the events on 4 March in Locharbriggs - where it is based - and nearby Dumfries.
Dumfries and Galloway Council's licensing panel is being asked to issue permission for both events.
However, a parish priest in Dumfries has objected to the march saying it has "no reason whatsoever, historical or otherwise" to be held in the town.
The council has to be consulted by anyone wishing to hold any public procession.
The Locharbriggs Loyal Orange Lodge wants to stage one march through Dumfries at 15:30 on Saturday 4 March lasting for about 45 minutes.
It would follow a route along New Abbey Road, Laurieknowe, Galloway Street, Buccleuch Street, High Street, Shakespeare Street and Nith Place before ending at the Dock Park.
The second procession - starting at 18:00 - would take place for about half an hour through Locharbriggs.
It has been estimated that 75 people would take part in the Dumfries procession and 50 people in the Locharbriggs one.
Fr Jim Hayes of St Teresa's RC church has made a representation against the marches.
He said he was "dismayed" to see a procession being brought into the "heart of Dumfries".
He said he hoped marches of this kind by "any group of any persuasion" would "disappear from Scottish life altogether".

The 22-year-old made two senior appearances while at Craven Cottage and had loan spells at Shrewsbury and Portsmouth.
After leaving Fulham, he spent the 2015-16 season at Dagenham & Redbridge, playing 38 times in League Two.
Last season, Passley made 13 appearances in National League South for Whitehawk.
Find all the latest football transfers on our dedicated page.

Archaeologists used a range of techniques to determine the age, diet and health of livestock raised at two sites in Orkney.
One of the sites, Earl's Bu, appears to have been organised and managed as a large estate farm.
It may have been run this way from an earlier time than previously thought.
The study, which involved excavated animal remains and the examination of other historical evidence, could give fresh insights into ancient farming elsewhere.
The University of the Highlands and Islands, which did the research, said these other places could include other Scottish islands and Scandinavia.
The archaeologists said the good condition of the livestock at Earl's Bu gave an "overall impression" of an organised system of pastoral farming on the land.
Herds of sheep and cattle were pastured and fattened up in established fields or upland areas, they said.
This suggests that an elite group of people controlled the land in the form of manorial estates from the Viking period onwards.
The archaeologists said that this challenged the view of agriculture in the 9th to 11th Century being "dominated by free peasant farmers with varying degrees of obligation to chieftains and larger farmers".
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