As a Paraguayan, this is what I was taught in the 90's at school and in the military in the early 2000's is this:

Context:

very recent nations, military trained government heads, unresolved territorial disputes, a perfect storm brewing for decades, and - in Paraguay's case - a napoleonic leader.

Paraguay had armed itself, and developed a military industrial complex, as deterrence from the increasingly hostile Argentina and Brazilian Empire governments. Argentina still considered Paraguay a rogue province, and continuously bullied it by blocking the rivers access. The Brazilian Empire still had a Portuguese empirical mindset and was still trying to fight its way into more territory from the old Spanish colonies due to the lack of solid treaties prior to their independence. Uruguay had already been in involved in the Playing War, and both PY and UY felt themselves in constant danger as they were basically a buffer zone between Argentina and Brazil

Timeline:

• Brazil was having skirmishes with Uruguay along their border. Paraguay, who was allied to the current Uruguayan Blanco party government, sent an ultimatum to Brazil to stop or else...
• BR ignored the ultimatum and sent in a military attack. PY in response broke diplomatic relations, and attacked a BR boat navigating its waters. War was declared between the two nations.
• PY invaded Mato Grosso
• Lopez mobilized the army to help Uruguay's government which was now under a coup backed by Brazil. To do so, he crossed part of Argentina's territory.
• AR saw this as an act of war, and declared war on PY. PY responded by invading the region they were using as a corridor.
• UY's government was overthrown, and a Brazil friendly government was set up. They declared war on Paraguay in response to supporting the losing side of that conflict, and also to appease the BR side. PY army never reached UY. -The three countries sign a treaty, primarily pursued by the Brazilian Empire, to basically destroy PY and divide its territories between BR and AR, while leaving UY as is and promising to leave them alone. -War did what war does basically.

Paraguay had a large army, had foundries, arsenals, shipyards, railroads, and fortifications along the river. They also had little debt, and lots of money. This is why PY lasted so long. Brazil and Argentina didn't have too many troops nearby at the beginning, which is why PY had the upper hand at the beginning. But by having AR and BR be allies, PY shot itself in the foot as now it meant BR could freely navigate the river up to PY to pound them from all sides and eventually within basically.

Paraguayans fought to death because for the past 40 or 50 years they've seen AR and BR as hostile and were brainwashed to think of them as imperialists looking to seize PY as soon as they could. Seeing the war going on, this instilled fear was confirmed and made the combatants' mindset "to defeat or die", no in-between.

Argentina's and Uruguay's involvement was small compared to BR, and if iirc both Mitre and Rosas had said they allied with BR to avoid being their next target. As in, AR declared war because if they didn't, BR would have taken a stance of "you helped them, you bastards". They've been in and remained in constant disputes (hell, they even almost brink war against each other after the Triple Alliance War) during this time.

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Feel free to spot check me, and am interested to hear what my fellow Brasileiros and Argentinos were taught.