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In a fantasy world I have planned a historical event was the "Bishops War" in which the bishops of multiple large parishes waged war with one another with large armies of warrior monks. But I've run into a pickle: trying to justify these bishops having armies in the first place.

The society is comparable to Medevial Europe in the High Period with a quasi catholic church. These monastic armies would be different from orders like the Templars or Teutons due to being under the direct control of a bishop and having a home cathedral. They would most likely be organized at a parish level.

I think you're only having this problem because you're too focused on Christian history and, as many others have commented, even that is full of religious leaders with big armies. The history of East Asia is absolutely STUFFED with examples of militant religious organizations, with Shaolin being the most notable example. If you want more current examples, religous leaders (mostly muslim, but not all) all over the third world have their own militias.

The bottom line is that all you need is a perceived threat to the established religious tradition in a culture, and you immediately get a LOT of fear and anger that can be mobilized to create a group of people willing to use violence to enforce religious orthodoxy. Secular politicians often welcome this because it redirects social tensions and anger away from the government and towards (usually) social minorities of all kinds.

The BEST example of the sort of warring religious organizations is the Sunni/Shi'ia split in Islam. That schism happened pretty much the next day after Mohammed died, and they've been killing each other ever since.

TLDR: All you need is to give the Bishops an excuse to make their parishioners believe there is a wide-scale threat to their cultural norms. This works even better if the threat can ALSO be portrayed as a threat to economic prosperity.

Once the Bishops HAVE those armies, they'll find excuses to use them against each other. They'll use morality and religious orthodoxy as camouflage for struggles over wealth and power, just like any secular leader.

Prince Bishops were a real thing in medieval and early modern Europe, mostly in the Germanies and Italy. The bishop of Rome, whom the western Catholics call the Pope and we eastern Orthodox call the Patriarch of the West, is still, in our days, a sovereign prince and still commands armed forces. Prince bishops were bishop and they were princes; as princes, they had armies and, quite obviously, participated in wars.

Swiss Guard swearing-in ceremony at the Paul VI Audience Hall. The Pontifical Swiss Guard is an armed force maintained by the bishop of Rome, aka the Pope for western Catholics and the Patriarch of the West for eastern Orthodox. Photograph by Paul Ronga, available on Wikimedia under the CC BY-SA 3.0 license.

Three of the Prince Bishops in the Germanies (the bishops of Cologne, Mainz and Trier) were even Electors of the Holy Roman Empire, with the titles of Princes Elector and Arch-Chancellors of, respectively, Italy, Germany and Burgundy.

As for specifically a monastic army, those existed too in real history. The most famous of them are of course the Teutonic Order and the Knights Hospitaller, both which ruled over actual states -- the Teutonic Knights in historical Prussia, and the Knights of Malta in, obviously, Malta. And yes, those monastic states had cathedrals...

Warrior monks are monks first.

Medievals did not keep standing armies. A bunch of layabout soldiers, waiting to cause trouble, mouths open for food? No way - put those men to work and then get them soldiered up when trouble calls. Maybe they can practice archery on the weekend.

So too your monks. They are just monks: making delicious beer and cheese, tending fields, green Chartreuse; monkish things. Prayers. Kung fu. The stuff monks do. Then when it is warring time, the monks gird their loins and go to war.

The monks that are too old or sick to fight keep the chickens fed and the sourdough starter alive back home.

History has already answered your question, during the Münster rebellion

The Münster rebellion was an attempt by radical Anabaptists to establish a communal sectarian government in the German city of Münster.

The city was under Anabaptist rule from February 1534, when the city hall was seized and Bernhard Knipperdolling installed as mayor, until its fall in June 1535. It was Melchior Hoffman, who initiated adult baptism in Strasbourg in 1530, and his line of eschatological Anabaptism, that helped lay the foundations for the events of 1534–35 in Münster.

The city was then besieged by Franz von Waldeck, its expelled bishop.

Long story short: if you want to exercise power, you need an army to enforce it. If your bishops have also secular power (which wasn't uncommon in those times) they ought to have an army at their order.

The bishops have both temporal and spiritual powers.

You may think of bishops as clergy, but the church was a major landowner. In addition to that, some bishops were ex officio nobles. Consider the Prince-Electors of the Holy Roman Empire. Four of them were secular, three were ecclesiastical.

In their role as nobles, the bishops and archbishops could collect taxes, hold court, and even wage war.

Like always, it is good to take a step back and look at the bigger picture.

For all of humanity's history, the most important thing is power. Since personal power without magic or chi or handwavium is rather limited, power comes from control. Control of masses in the middle ages, since cold weapons were limited force multipliers.

Control has many ways. Money, faith, custom, loyalty, force...

The reason the bishops (religious leaders) didn't have standing armies (with exceptions mentioned by AlexP's answer) because they had the masses. It was low quality but high quantity.

The Pope excommunicating someone was a serious flexing of their control.

Naturally, rulers wished to offset this threat to their authority, so they forbade, restricted religious figures to own or train a sizable army. Guards were mostly ok.

If you want bishops with standing armies, you have to change your worlds setting in this regard.

Options:

1) The king has a strong religious background and backing so he encouraged the bishop to train armies to strengthen himself against other threats to his crown.

2) The crown is weak and unsuited for ruling, forced to allow the bishops more control. They will naturally want to strenghten themselves, not just by gaining rights but by more ... direct means.

3) It is all a ploy by the ruler to weaken the church. It allows them to gather armies then foster mistrust and greed among them. A conflict between bishops, for the Pope position would weaken their religious control, as they are exposed as greedy, maybe sold their souls to the devil, not caring for their flock, going against god's teaching...

The options are virtually endless...

In Medieval times many bishops became secular nobles and thus vassals of higher lords. So that meant that in times of war they were required to provide warriors; knights, men at arms, mercenaries, and armed peasants, etc., to their overlord's army and and sometimes lead them in battle.

Secular nobles, including the ones who were also bishops and archbishops, also often fought other nobles in private wars or civil wars.

This was especially the case in the Holy Roman Empire. I remember a biography of King Henry III's brother Richard, Earl of Cornwall, who was elected King of the Romans, quoted Richard as saying that the Bishops in Germany were very warlike.

Of course bishops would call upon their secular vassals to provide secular warriors, or money to hire mercenaries, when the bishops went to war. It was not normal for a bishop to order monks, priests, cathedral canons, etc. to go to war, since most of them didn't practice fighting skills during peacetime.

Dohn Joe's answer mentions The https://en.wikipedia.org/wiki/Goslar_Precedence_Dispute1

It was an armed fight over precedence between clergy and laymen loyal to the Abbot of Fulda and the Bishop of Hildesheim, resulting in bloodshed and death in and around the Church of St. Simon and St. Jude in Goslar. Count Egbert of Brunswick led the fighters for Hildesheim and so probably commanded secular warriors. The fighters refused to listen to the 12-year-old King of the Romans Henry IV commanding them to stop and he had to leave the church for his safety.

There were a number of monasteries in the Holy Roman Empire that acquired lordships or counties and became fiefs, and later states, of the Empire. Thus a number of medieval abbots lead their troops in various battles.

In the early modern era when the Holy Roman Empire declared war each state of the empire was supposed to deliver a specified number of soldiers to the imperial army. That included the bishoprics and archbishoprics and those monasteries that were states - including the monasteries ruled by abbesses. Some of the smaller states were required to deliver fractional soldiers - twelve and half men, for example. I suppose that two such states would each agree to pay half the expenses of a soldier.

There were a number of monastic orders that include monastic knights - the famous one are the Knights Templar, the Teutonic Knights, and the Knights Hospitaler (Knights of Malta) but there were also knightly orders in Spain and Portugal.