Broadly speaking yes.

While not every party had it's own armed organization, the ones that took part in the violent street-fighting of post-WW1 Germany can be atleast classified by general political alignment.

Firstly, there was the Freikorps, also known as the Black Reichswehr. Organized by what remained of the german general staff, the freikorps were supposed to act as the german government's anti-uprising force 'outside' the official german army strength permitted by the treaty of Versailles. Made up of nationalistically-minded front veterans, the freikorps rather quickly aligned with the right wing.

Then there was the Nazi-organized Sturmabteilung, which is the paramilitary organization you saw in the documentary.

Then there was the Stahlhelm. Officially a veterans' organization, the Stahlhelm represented the interests of german nationalist conservatives and monarchists. The Stahlhelm resisted left-wing activism of all sorts however and whereever they could, which led them in bed with the Nazis. Eventually they were incorporated into the Nazi SA.

Beside these there was a number of smaller organizations such as the Bund Oberland and the Bund Reichskriegsflagge etc. Something worth noting is that the Nazis created an umbrella organization called the Kampfbund to coordinate the various right-wing fighting organizations to their own ends.

In the political center we have the Reichsbanner SRG, which was closely aligned to the Social democratic party and dedicated to defending the interests of the Weimar government. This organization was banned in 1933 when the nazis seized power.

Also in the center, the Eiserne Front, which was formed by liberals and democratic trade union actives specifically to oppose monarchism, fascism and communism alike in favour of liberal democracy.

And on the political left we have a big federation of local fighters' leagues, antifascist actions and soldiers' councils called the Rotfrontkämpferbund, usually called just the Rotfront among friends. The Rotfront central committee took orders from the german communist party and thus, Moscow. Officially banned by the social democratic government in 1929 as too extreme. This is still somewhat of a political hot button in the german left wing, as I understand.

My source for this post is The Coming of the Third Reich by Richard J. Evans

Yes, it was normal in the sense that German political parties of the Weimar era often had armed factions. What was unusual about this arrangement was how formal it was, meaning that the paramilitaries were official parts of the parties and not just allies or ideologically aligned.

(Pardon me if I get long-winded, but this is close to my research specialty.)

We can look at the background in two ways: first, as part of factional violence in democratic politics across the transatlantic world, especially in urban areas and big cities. Second, as an outgrowth of specifically German political behavior, both dating back to the 19th century and the specifically the early Weimar period.

To take the first, 19th century urban politics often featured factional violence and the participation of armed gangs. These were usually allied with political parties, but kept a formal separateness. Think for example of how Sinn Fein was the political party of Irish nationalism, but the IRA was the paramilitary. The two did not always see eye to eye, and formally split on several occasions. Everyone knew the two were connected and cooperated, but they were still formally separate entities. Similarly, democratic politics in Italian city states featured factional battles and armed conflict dating back to the Renaissance, which mixed family/tribal and neighborhood allegiances. Urban machine politics of 19th century United States northern cities often featured an alliance between the political party, the machine faction, and the armed gangs. So in New York's case: the Democratic Party, the Tammany Hall machine, and various factions like the Bowrey Boys, the Dead Rabbits, and other neighborhood toughs. (To see it on film, check Martin Scorsese's amazing "Gangs of New York," based on Herbert Asbury's 1927 history of the same title.)

So on the one hand, gang violence and democratic politics are not as estranged as we might think.

19th century German political traditions featured political factions as well, however, they were not allied with political parties per se until very late in the century. Historians of Weimar paramilitaries often look to the student dueling fraternities (Burschenschaften) of the early 19th century, which combined lightly violent swordfighting rituals with political debate and activism. These were very important in the 1830s debates over nationalism, and then the 1848 revolution. We then trace political factions in the imperial period to infamous nationalist clubs like the Pan-German League, the Navy League, and others who advocated imperialist and nationalist policies. They were political, but not party-political, and they were not armed. But a lot of historians of the Weimar paramilitaries look to them as precedents, because there are generational connections between those who were in these groups, and their sons or grandsons in the paramilitaries.

The end of WWI and the violent birth of the Weimar Republic created a new dynamic for political clubs, firstly because of troops of armed veterans took it upon themselves to put down civil unrest and defend border zones. These Free Corps (Freikorps) were more like warlord bands, usually organized around a charismatic leader who had access to weapons. The early Republic, lacking a coherent military force of its own, allowed them to put down far-left attempts at revolution. However, the Freikorps were not loyal to any party, and they were particularly treacherous against the democratic center. Same thing with the Stahlhelm, the right-wing veterans association that backed anti-democratic rightist parties, but never a single party.

These Freikorps are the direct predecessors of the nazi stormtroopers (SA), in fact one of the books about them is titled "Hitler's Heralds." But it's really the SA that changed the dynamic for good. Hitler founded the group in 1921, specifically for the alleged purpose of protecting party meetings from the disruptions of opponents. This is the connection to the 19th century German political clubs - political debates were held in beer halls, taverns, and other public spaces, and it was accepted that the audience would engage in some amount of catcalling and challenging the speakers. But after WWI the tone of these turned more combative, and people got into fights more. So Hitler formed the SA -- allegedly! -- to protect against this behavior, and in fact the founding myth of the SA as he told in Mein Kampf describes their "birth in blood" of turning back an enemy attack on a Nazi speech in Munich. However, note that I say allegedly, because as the history of these paramilitaries made clear, they were the ones attacking others far more often than vice versa.

The Nazi creation of the SA therefore started a sort of political arms race in which the other major factions needed to have their own domestic military forces. The SA had been banned in 1923 for its role in Hitler's Beer Hall Putsch, but when it returned in 1924 the others responded. The democratic center created the Reichsbanner in Feb 1924, allied with the Social Democratic Party, the Catholic Center Party, and other democratic centrist parties. They swore to defend the republic against authoritarian enemies from right and left. The Communists created their Red Front Fighting Brigade in July 1924, to attack the center and right. The Reichsbanner was the largest in terms of numbers, but the Red Front and SA were the most aggressive and notorious. Which is very fitting for this question -- more people can support the center, but this form of politics gives advantages to extremist factions.

By the late 1920s, the SA began to out-compete the other rightist paramilitaries, and eventually began absorbing them en masse when a group would dissolve in favor of its members joining as a pre-organized SA troop. (See Andrew Wackerfuss' "Stormtrooper Families" for examples of this strategy in Hamburg; it allowed the SA to mobilize pre-existing community relationships, or metaphorical "families," to grow its ranks and eliminate competitors.)

I'll conclude with the implicit question behind the question: what did the parties get out of this? What role did the paramilitaries play? Well, physical defense of their own spaces, and in the Nazi/Communist case attacks against their enemies' spaces. But also they served to create the a public perception of mass support, both in terms of numbers and financial backing. You could see the militia marching, in its uniforms, and it made concrete the assertion that a lot of people supported that party. (Never mind that in a lot of the Nazi marches, they had to bring people in from surrounding areas to pump up their numbers and make a better impression. But that too was an example of logistical strength and organization.) The SA also advertised itself as a social welfare organization, providing its unemployed members with food, clothes, lodging, etc. (Wackerfuss has a large section on this, as well as Daniel Siemens' "The SA," which is the best overview history of the group in English.)

The point of the SA, when you get down to it, was to increase the violence in the Weimar system. Start fights, break windows, disrupt political debate, attack people in the streets. Then the political wing would turn on a dime and decry "Why can't the republic keep order in the streets? Look how bad things are. Elect us and we will bring order." Never mind that they were often the ones behind the violence. It worked on enough people, especially the center-right that eventually cooperated with them to leverage them into power. The Nazis never got a majority of the vote, but they got close enough where they could take power, then change the rules to not lose it.

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SOURCES:

Herbert Asbury, "Gangs of New York: An Informal History of the Underworld" (1927)

Nigel Jones, "Hitler's Heralds: The Story of the Freikorps" (1987) -- a little out of date but still good

Dirk Schumann, "Political Violence in the Weimar Republic" (2009 updated edition)

Daniel Siemens, "The SA" (2018) -- the first full-length comprehensive English-language history of the SA. Currently the gold standard. His biography of Horst Wessel is also excellent.

Pamela Swett, "Neighbors and Enemies: The Culture of Radicalism in Berlin" (2004) -- superb street-level look at the factions and fighting in Berlin

Andrew Wackerfuss, "Stormtrooper Families" (2015) -- SA community, relationships, and street violence in Hamburg. Includes two detailed chapters on specific battles that show the dynamics of combat between factions.