The dichotomy between the relatively militaristic bent of the US and the total lack of military parades is somewhat strange. In many countries (including other democratic countries) it's normal for everything from tanks to ballistic missile launchers to get trotted out in the hundreds on national holidays, but the US simply doesn't do this. On the other hand, there is the weird tradition of warplanes flying over football games, so it's not like the US is averse to open displays of military hardware. I don't get it.
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.
Many military bases host regular events showing off their war assets. Around here (DC Region) there's the Joint Service Open House, a weekend-long event held at Andrews AFB where just about every military land/sea/air unit you can think of is on display. It's essentially a huge airshow with military aircraft, while a huge area of the tarmac is dedicated to static displays of stuff like tanks/boats/launchers/etc. It may not be a parade, but it's more or less a way for the military to show off their skills and fleets. The JSOH is a huge event for the region, but plenty of military bases host events like this all the time.
I think its worth pointing out that the flyovers you mention usually aren't just for show. They are part of training and simply coordinated so that the training occurs during the right time and place as events.
I don't know about this as the history of flyovers, but I think it is an important distinction in relation to your question because rolling tanks, APCs and missiles down streets isn't a part of regular training, while flyovers are both training and outreach and your question seems to be concerned about the show and outreach element.
