Reverend Stanley Gamble was speaking at the only Irish Republic Orange parade which takes place annually in County Donegal.
More than 12,000 marchers and 30 bands marched along 2 miles of country roads.
Rev Gamble stated: "British passports should be available to Irish citizens south of the border, just as Irish passports are available to British citizens north of the border. This would be a real and tangible sign of change and hope."
He said that he looked forward to the queen visiting the Republic, as has been reported.
Rev Gamble said: "Won't it be great to see the queen afforded that much anticipated and long overdue visit to Dublin.
"And I have no doubt that the great majority of Irish people will extend a warm and friendly welcome to the queen, just as they did to her grandfather King George V, 100 years ago next year."
Orangemen have paraded in Donegal since the early 1900s.