Grand secretary Drew Nelson has made history by becoming the first member of the organisation to address the Irish parliament.
Mr Nelson said Ireland would be a "poorer place" if the order's cultural heritage disappeared and asked for accommodation and tolerance.
He also suggested the Irish state had failed to look after Protestant communities in border counties as he claimed the British Government looked after Catholics in Northern Ireland.
Mr Nelson told senators in the upper house that one of the order's main goals was to improve north-south relations by holding parades in the Republic.
"There was one planned in Dublin a few years ago but it was unable to proceed," Mr Nelson said.
"Our members in the Republic would welcome the opportunity to hold a parade in their capital city."
About 20 Orange Order parades take place in the Republic every year but none in a major city.
The only attempt to hold a major demonstration in Dublin - the Love Ulster march in 2006 - was abandoned after hundreds of protestors rioted on the streets of Dublin.
Mr Nelson said the institution understands the challenges a parade in the Republic's capital would pose.
This institution and the bands which we support are the guardians of part of the intangible cultural heritage of not only Northern Ireland but also the Republic of Ireland.
On the issue of falling Protestant populations in Donegal, Cavan and Monaghan Mr Nelson said the dwindled numbers compared to growing Catholic communities north of the border.
"This of course begs the question as to which state looked after its minority better," said Mr Nelson.
He said many of them have spoken of their "fear of incurring the displeasure" of the State in any way - not necessarily a fear of violence.
The grand secretary, a key player for years in the Orange Order hierarchy, said it was important the Government is aware of the issue.
Mr Nelson also condemned recent sectarian attacks on Orange Order halls, which he described as the "demonisation" of the organisation by some members of the Republican movement.
He said resistance to parades including the annual Twelfth parades has a "corrosive effect" on Catholic-Protestant relations.
The grand secretary thanked Fianna Fáil leader Micheal Martin who invited him to the Irish Parliament, comparing the gesture to that of the Queen's visit to the Republic last year.
He told the Seanad: "Together let us resolve that no longer will the burden of history stand in the way of normalisation of relationships."
Mr Nelson responded to remarks from senators, including Sinn Fein's David Cullinane, through the chairman. The Orange Order does not speak directly to Sinn Féin.
Mr Cullinane claimed the historic Seanad speech gave him an opportunity to address a member of the lodge directly.
In his groundbreaking address, Mr Nelson appealed for the Irish Government to continue to support Protestant schools in Ireland.
He claimed communities, particularly those in the border counties, face cuts to Church of Ireland and other Protestant schools, which do not fall under state control.
"The Protestant community actually fears for its continued survival as a viable, self-sustaining community," said Mr Nelson.
"I appeal to you today to take whatever steps are within your power to address that issue and reassure our members living in the border counties."
Senator Martin McAleese, husband of former president Mary McAleese, told Mr Nelson of his fear as a Catholic child growing up in Loyalist east Belfast and watching Orange Order parades.
He said he had hated marching season and felt threatened as part of a minority community but recent cross-community cooperation between himself and Mr Nelson had helped him a develop "an appreciation" for the order's heritage.
Seanad chairman Paddy Burke said Mr Nelson's presence in the chamber was an example of progress made between the north and south.
He said he had little awareness of the Orange Order as a child having grown up in rural Ireland, and therefore believed the organisation to be "far removed" and "irrelevant" to Irish life.