For the United States, I can shed some light. The United States Army and other Armed Forces of the U.S. are, at a federal level, heavily restricted from operating as a cohesive force for any purpose (including parades) domestically.

One thing that's important to understand is how the U.S. Army is organized. While there is a Regular Army and an Army Reserve that are controlled, mostly, at a central, federal level. There is also a large National Guard that is, at least theoretically, the responsibility of each state to raise, maintain, and control, and which is the primary go-to for domestic army needs, not the Federal Army forces.

Indeed, the Founders of the U.S. were extremely wary, for obvious reasons, of standing armies of any kind, and especially of the chief executive (in this case, the President of the U.S.) using those armies on U.S. soil for improper purposes under the guise of protecting the people. As such, soon after the U.S. Constitution was passed, the Congress passed the Insurrection Act of 1807, which makes it very difficult for the President to use the army domestically, even in the event of an insurrection, rebellion, or foreign invasion, without going through a series of Congressional checks. Likewise, the Posse Comitatus Act, of 1878 and 1981, heavily limits the ability of the federal Armed Forces to be allowed to enforce state laws.

Together, along with other supporting laws, it is very, very difficult to allow the Army to operate in any capacity at all in a domestic way in the U.S. Parades that involve large gatherings of troops and marches would not necessarily be prohibited by any of this, but the troops that would be organized in this way would likely need to come from the various states' individual National Guard units, not the overall Federal Army. States are unlikely to want to send large numbers of their personnel to a single point in the country for a show of force, and the nature of these laws and the philosophy behind them should show that Americans, in general, are pretty put off and suspicious of parades of this nature, to start with.

That said, such gatherings were not unheard of in the old days. During the Civil War, when States were truly responsible for gathering their own Army units and there was no separate Federal force to speak of, there were parades and shows of force. The Army of the Potomac paraded down Pennsylvania Avenue, as seen in that photo, during the Grand Review of the Armies following the end of the war in 1865. In a funny story, during this parade, George Armstrong Custer, in typical fashion, actually paraded twice, just so he could soak up double the adulation from the adoring masses. That egoism would later serve him poorly.

At any rate, I've sort of rambled around, but the short answer to your question, at least with regard to the U.S. is: Americans are suspicious of domestic armies, and put lots of laws to keep them out of domestic operations in place. That, along with the state-focused organization of the modern U.S. Army makes parades hard. However, they aren't unheard of in the past, even here.

Actually military parades are much more common than you may think, and are certainly not circumscribed to autocratic states. Even though the USSR collapsed, Russia continues to hold a parade to commemorate winning WWII (The Great Patriotic War). Here is a link for the 2013 celebration.

You already mentioned France, but Spain and Italy also have military parades. Those are the examples than come to mind right now, but there are certainly more. Check this Wiki page which list the Armed Forces Day of different counties, since military parades usually are usually held for that day.

There may be other answers to this question as well, but in large part it has to due with regime legitimacy. Remember that democratic regimes in general derive their legitimacy from elections. In the modern day, elections have taken on such great meaning that in many authoritarian states (see North Korea), "elections" are still "held" just to prove regime legitimacy. The fact that elections in democracies are patently more fair and competitive means that democratic regimes generally don't need to rely on alternative sources of legitimacy, at least at first. People elect democratic leaders and give them some leeway. Later, if the regime's legitimacy has been eroded (say because of corruption, poor economic performance, etc.) then a new regime can be elected.

In autocratic regimes, citizens know that elections are generally not competitive. Without fair elections to provide legitimacy, regimes often turn to other sources of legitimacy. Some of these stem from a regime's ideology. For instance, Baathists may claim legitimacy from policies which support Islamic culture. A common way for any authoritarian regime to prove its legitimacy is to show its military strength. Many of these regimes have arisen in response to some external threat (say the CPC) or in order to export their belief systems elsewhere (say the USSR). No matter what, a strong military shows that the regime is capable of defending its citizens and ostensibly also proves a certain level of economic success (you can't design and pay for missiles without a basic industrial economy, after all). In many regimes, autocrats have a special fondness for military parades because their supporters predominantly reside within the military. In these cases, military parades can also please the generals on whom leaders rely by identifying the military as a prestigious source of national pride.

The Autocratic states of the early 20th century placed a lot of value upon their military might, and as such had regular military parades to showcase that might.

There is a misconception in associating the parade with them alone, however. These regimes may have used parades to the greatest propaganda effect, but many democratic nations have parades as well.

The military parade has been a longstanding tradition in European countries, practically since the creation of national regular standing armies. Until the 20th century, parade maneuvers were actually a practical part of training soldiers. The parade formations you see today, while certainly not intended for combat even in the era of horse and muskets, are very similar to the kind of formations that the soldiers of the day would have had to fight in: Lines and columns of men who would have to move in a very tight formation. Practice on the parade ground fostered company cohesion on the battlefield. (And it payed dividends - at the battle of Balaclava in the Crimean War, parade style precision allowed the British Highlanders to concentrate fire on a single point from their entire line without worrying about hitting any of their fellows. This concentration stopped a mass Russian cavalry charge)

Even today, parade drill is a part of military training in most nations, as it still helps with discipline and cohesion (though not in the immediately practical way it used to).

The parade also became a tool for raising the desire to fight at home - the soldiers marching in lock step, wearing stunning uniforms, with flags waving in the wind coming to town was a big to do back in the day. (In the 18th and 19th centuries, it was also a way for commanders, who were often times members of the aristocracy, to show off and gain prestige)

Also, Who says the US doesn't have military parades?

Japan has one too.

Even South Korea has a military parade.

And for you Canadians, there apparently was one in Toronto fairly recently.

A simple youtube search of "Military Parade" will get you quite a few results you might not have expected.

I realize it's not on the same level but I think it's worth a mention. The US does have the Blue Angels which is a full time, world-travelling show designed to show off military prowess.

The mission of the United States Navy Flight Demonstration Squadron is to showcase the pride and professionalism of the United States Navy and Marine Corps by inspiring a culture of excellence and service to country through flight demonstrations and community outreach. Source