Well, the rise of Paraguay depended hugely on the lucky timing of their self-assertion in the Southern Cone. They managed to confront then Argentina at a time of critical division, resulting in its collapse. At that point, with friendly Chile, Entre Rios and Tucuman, Paraguay could go on to vassalize Uruguay having only Brazil to worry about. And the Brazilian army was in pitiful state at that point. The political instability and internal division that followed that defeat allowed Paraguay to profit and support the Gaucho Republic of Rio Grande in its successful Second war of Independence. The fall of the monarchy crippled Brazil permanently and permitted Paraguay to consolidate its dominant position.
When Brazil, Buenos Aires and even Tucuman recovered, they did so in a Paraguay-dominated context and had actually no desire to challenge that as they profited from the regional stability that Asuncion guaranteed (except for the Partition of Bolivia of course, but that was a reaction to Chilean action for the most part).
However, if Imperial Brazil and Argentina could keep themselves together, they'd outmatch Paraguay so much it's not even funny. The country could be sandwiched between far stronger neighbours and be screwed. Now, both states collapsed mainly as a consequence of well-timed Paraguayan intervention, but it did not have to be necessarily the case. If Paraguay had antagonised both countries at the same time, even in their sorry state, it would have probably lacked the resources and population to emerge to dominance.
Think of the Swedish empire.