I have to go off to work, so I can't continue researching, but I do want to give you what I've found so far. The lack of military parades appears to be a relatively recent habit.
For example: In 1864, the New York Times declared the military parade to be "the finest feature of the celebration" of Independence Day. After the American Civil War, military parades declined. Gen. Ulysses Grant was famously against them -- he declined parades in his honor while visiting military installations after the war.
In 1868, the New York Times observed a notable lack of military parades in celebration of Washington's birthday: "The absence of any military parade, with the exception of that of a solitary regiment, left a void in the celebration of the day to which our citizens have not been accustomed. It is true that business was generally suspended, but the absence of any special demonstrations was noticeable, and hardly to be accounted for."
Military parades seem to fall out of fashion unless there's a war. They took place during the Spanish-American War, the First World War, and the Second World War. Here's a particularly good photo of an M2A3 Stuart tank during a parade in front of the U.S. Capitol during World War II.
I'll conclude with an 1866 letter written by a veteran which, I think, explains the American reluctance for military parades. It's a strongly anti-war letter and states in part: "... I have no admiration for the military profession, no desire that war should continue, and nothing but contempt for what are justly thought the mere pomp and glitter of military parade. But, alas! for our poor human nature, wars must come, and military pomp will attend them."
I hope someone else will take up the research here, but it seems to me that in the United States, military parades have been supplanted by such things as air shows, Fleet Week, Veterans Day, Independence Day, flyovers, and other things that don't specifically resemble your prototypical military parade.