THE TROUBLES in Ireland, which began in the reign of Queen Elizabeth I and continue into the reign of Queen Elizabeth II, have about them a quality of endlessness. Resolution is ever at hand, never achieved.

It is the same with the mirrored American version of the world's bitterest and most enduring quarrel. For the past seven years, a struggle to part Irish-Americans from their sentimental attachment to the Provisional IRA, the "provos," has been waged by the tribal chieftains. The Four Horsemen, former New York Gov. Hugh Carey, Sens. Daniel Patrick Moynihan and Edward M. Kennedy, and House Speaker Thomas P. O'Neill Jr., began in 1977 to tell their own in annual St. Patrick's Day statements that Irish violence is not the answer to English injustice in Ulster.

The idea that the "provos" were thugs, not the descendants of the martyrs of the Easter Rising immortalized by W. B. Yeats, seemed to be gaining ground. At every opportunity, the Horsemen, who were eventually folded into a larger, bipartisan Friends of Ireland pointed out the horror and futility of terrorism.

They seemed to be making headway, even against Rep. Mario Biaggi's mischievous ad hoc House committee on Ireland -- Biaggi once proposed to invite the gunmen here to give their side of the story. They held their own during the wrenching days of the hunger strike and the series of prison deaths that set bells to tolling in Irish minds. The election of Garrett Fitzgerald, a humane and scholarly man who has made Irish peace and unity his first purpose in life, promised renewal of the promising Anglo-Irish talks.

All these hopes went spinning on Feb. 9, when the Ancient Order of Hibernians, organizer of New York's parade, elected as grand marshal 82-year old Michael Flannery, an IRA man in his youth and a cofounder of the pro-IRA Irish Northern Aid committee (Noraid). Flannery, who has just been acquitted of gun-running charges, promised to turn what is supposed to be a great day for the Irish into "a pro-IRA parade."

Once again, as so often happens in Ireland, the rock had rolled down to the bottom of the hill.

The Fitzgerald government was infuriated. Within five minutes of the desperately embarrassing announcement of Flannery's election, it withdrew its official support. Moynihan, who was roundly denounced for it, said he would boycott the parade. Carey said the same.

But New York's most conspicuous non- Irish politicians, who expressed an irrational desire to steer clear of the politics of the situation, namely Gov. Mario Cuomo, Sen. Alfonse D'Amato, Mayor Edward J. Koch and inevitably, Biaggi, opted to march down Fifth Avenue with the extremists.

The exploitation of St. Patrick strikes serious Irishmen as nothing less than obscene.

Tadgh O'Sullivan, the Irish ambassador, said, "We deplore the use of the healing and unifying symbol of Ireland's National Patron for such bitterly divisive and destructive ends." St. Patrick, of course, was the victim of the terrorists of his day, but unlike the typical Gael, harbored no bitterness against his oppressors.

As usual, there is plenty of rancor to go around. Much is applied to the president of the Hibernians, Joseph Roche, a Baltimore insurance man who, as a moderate candidate for leadership of the all-male, all-secret society, enjoyed the unofficial support of the Dublin government. But he showed his true colors last month when he accepted an invitation from Noraid to be guest of honor at its annual banquet. His participation in the choice of Flannery has led to charges of "treachery" by his erstwhile sponsors.

The situation seems to be nothing less than excrutiating for Terence Cardinal Cooke, the Roman Catholic archbishop of New York.

His Eminence has issued three press releases so far, none of which answer the most important question. Will he, as honorary grand marshal of the parade, greet the little pro-IRA grand marshal on the steps of St. Patrick's Cathedral and give Noraid what it wants most, a picture of a prince of the church endorsing an advocate of terrorism?

In his first statement, the cardinal, who seems torn between certain generous members of his flock and the principle of non-violence, vowed neutrality. In his second, in voicing his sympathy for the persecuted Catholics of Northern Ireland, he sounded so anti-British that his press release rattled the teacups at 10 Downing Street.

In a third, he expressed a curious hope that he might achieve some sort of reconciliation between the warring factions. How he could bring this about without actually asking Flannery to step aside is not clear. Peace and harmony are not Irish exports.