I'm currently reading The Chaco War (Farcau), and I've come across something interesting; Argentina and Brazil were prepared to partition Paraguay along the Rio Paraguay and Rio Parana after the War of the Triple Alliance, but Bolivia got word of this plan and nixed it, as they had claims over the Chaco up to the Paraguay. What if this secret plan had been kept better, and Paraguay partitioned in 1870?
Well, the "partition" of Paraguay didn't mean the extinction of the country (although apparently Mitre really said to the British ambassador in Argentina that he didn't to make an agreement with Brazil over the preservation of Paraguay because the "situation" could change in the future). The secret Treaty of Alliance, made by Elizalde and the Viscount of Rio Branco, included a clause that stated that Brazil wouldn't oppose Argentina taking the all lands on the right side of the Paraguay River until the Brazilian border. However, when the news of this agreement were received by the Parliament in Rio the MPs reacted strongly against it, as it would give Argentina a direct connection with Mato Grosso, and it could mean a future annexation of Paraguay (despite the fact they were our allies, the Argentines were still seen as "The Enemy" by the majority of politicians, while Paraguay was only an annoyance).
In the end, this clause remained in the limbo, as even the Ministry of Foreign Affairs were against it. The final relatory about the treaty explained that if Argentina didn't accept to reduce their claims over the Chaco then Brazil should extend the territory that should be annexed too, expanding it from the border on the Igurei River to the Iparrregaçu or the Aquidabó, in order to make Paraguay "a more limited prey to a future Argentine absorption".
So, basically, if the "partition" of Paraguay happens it would mean an increased tension between Argentina and Brazil, even greater if anyone tries to take the rest of Paraguayan territory. Also, it would increase tensions between Argentina and Bolivia over the Chaco.
Just to show the alternative claims of Brazil if Argentina had got the Chaco, here is a map showing the the Aquidabó River (1), the Iparregaçu (2), and the Igurei (3).