Some level of parity helps but it is not required.
The Yankees and Red Sox had a fierce rivalry that lasted about thirty years (it really began in earnest in the mid-70s) before the Red Sox achieved anything resembling parity in the mid-Oughts.
The teams' on-field rivalry really began in the DiMaggio/Ted Williams days. The highlight of that period was the 1949 season, in which Joe D. missed the first several months of the season with a foot injury. The teams were neck-and-neck all year, with the Bosox pulling ahead in the final week.
The Yanks and Red Sox were scheduled to end the season with two games against one another; the series began with the Bosox in first place by one game. The Yankees won the first game, which left the teams tied with a game to go, thereby making the final game of the season a
de facto playoff game. (Which the Yankees won.)
The rivalry faded somewhat after the retirement of Joe D. (whose brother Dom had played for the Red Sox and was on the 1949 team), and even further after Williams's career ended. The Red Sox didn't become serious yearly contenders again until the Yankees had entered their "lean years" after 1964.
Not until the mid-70s could the two teams battle each other again as perennial powerhouses; and at that point the rivalry flared anew to become baseball's fiercest. But it certainly didn't start there.
To answer the original question: the games against Toronto were the chippiest. The one against D.C. United was the most satisfying. Eventually the games against the Galaxy will have significance. But I cannot overlook the Red Bulls. That's *the* rivalry.
By the way, the setting on this board that changes N.Y.R.B. to NJRB is silly. There should be no issue with calling other teams by the names that they actually have, regardless of anyone's opinion of any particular name.
Furthermore, while I dislike the Red Bulls, I don't begrudge them the use of the name "New York". Indeed, the fact that they use the "New York" name is a tribute to New York City, in that it declares loudly that places in northern New Jersey derive their significance from their proximity to New York. We should note that Negro League teams called the New York Black Yankees and the New York Cubans played at a stadium in Paterson. (Incidentally, Paterson's Hinchliffe Stadium is the last Negro League park that's still standing.)
I have always been an arrogant New Yorker who unapologetically sneers at New Jersey. But, to be honest, Newark/Harrison is not one of the places that I sneer at. I like Newark a great deal; and I love the great cities of transit-rich Hudson County, such as Jersey City and the eminently bikeable Hoboken. A New Yorker can feel very much at home in those interesting cities, unlike in the awful nowheresville suburbs that make up so much of Jersey.
So my beefs with the Red Bulls are 1) that they retain the stink of early MLS; 2) that they carry a corporate name. But the fact that they use the locality name "New York" doesn't bother me in the least. It is only natural for a team that plays in one of North Jersey's large cities to identify with New York, as those cities are part of the great megalopolis of which we are the centre. This board should cease the practice of altering the correct abbreviation of the Red Bulls' name. Such pettiness is beneath us.