Complex sentence: Argentina stayed neutral and only declared war on Paraguay when it invaded Corrientes province after Mitre rejected the request Solano made to use Argentinean territory to move his troops to fight in Uruguay against Brazil.
Split into simpler sentences: Argentina stayed neutral in this, and that country only declared war on Paraguay when it invaded the Corrientes Province of Argentina. This occured when Mitre rejected the request that Solano made to use Argentinean territory to move his troops to fight in Uruguay against Brazil.