Although this is the "second" character to take the name El Papagayo within the time frame of the DC Universe , he actually pre-dates the Jonah Hex villain, first appearing way back in 1949 and left in obscurity before being revived years later in the modern era by writer Grant Morrison.
The original El Papagayo was the leader of an infamous gang of bandits in 19th Century Mexico who undertook such major crimes as looting banks and army posts. His name means "The Parrot" in Spanish and he always carried his Parrot, Pepito with him. El Papagayo ultimately intended to overthrow the Mexican government and establish a dictatorship.
El Papagayo first met his nemesis Jonah Hex when Hex was hired to infiltrate El Papagayo's gang and learn his plans. Hex thought he had been hired by the U.S. Secret Service, but in fact he had been set up by an old enemy. When El Papagayo discovered that the guns Hex had sold him had no firing pins, he turned against Hex, and the two clashed several times thereafter. Eventually El Papagayo got his revenge by slitting the throat of Hex's mother and Hex eventually clipped Papagayo's wings permanently.
Second El Papagayo
A later El Papagayo, with no apparent connection to the first,held a small South American country at bay with his cartel early in Batman and Robin's careers. El Papagayo was portrayed as a violent man who killed if his pet parrot Toto was not amused.
The president of that country contacted Batman and Robin to train a Bat-Hombre to confront and defeat Papagayo. Unbeknownst to Batman, Robin and the president, the man who they trained to be Bat-Hombre, Luis Peraldo, was secretly working for El Papagayo. When Bat-Hombre clandestinely met with Papagayo, Toto scratched Bat-Hombre's costume. When Batman and Robin saw this, they were able to deduce that Bat-Hombre was working for Papagayo and imprisoned him .
Rather than crush the president's dream, Batman temporarily took the mantle of Bat-Hombre to defeat El Papagayo. After a fight with the escaped Bat-Hombre atop a cliff, where Bat-Hombre fell to his death, Batman and Robin were able to defeat Papagayo by feeding Toto curare and knocking him unconscious. With the parrot knocked out, Batman used ventriloquism to "voice" the parrot and convince Papagayo to give the two whips under the guise of whipping one another to death. They were able to defeat the gang with the very whips they had given them.
Years after that confrontation with Papagayo, he resurfaced in Argentina. When Batman and El Gaucho confronted him, he and his gang were torturing an Argentinian hero called El Cimarron after it was found out he was attempting to infiltrate them as Black Miguel. Papagayo was in charge of a weapons facility that was manufacturing super weapons, in particular explosive blue scorpion used by Scorpiana. When Batman and the Gaucho attacked the gang, Papagayo attempted to get away in a hot air balloon only for Batman and Gaucho to follow. Papagayo was stung by one of Scorpiana's scorpions before they could extract who he was working for from him, and he fell to his death. Toto gave the two a hint as to his owner's employer who is Oroboro.