I have seen them in pet shops and there they were called Brazilian Cardinals.
The bird is commonly called the Red-crested or Brazilian Cardinal. It belongs to the family of tanagers and is actually not closely related to the cardinals and specifically the red cardinal found in North America. This bird is also NOT a woodpecker.
This species can be found mainly in northern Argentina, Bolivia, Paraguay, Uruguay, Brazil's Rio Grande do Sul and southern part of the Pantanal. The birds have been introduced to Hawaii and Puerto Rico. These are tropical and subtropical birds. I kept a few of these as aviary birds years ago.
Thanks for the nice introduction!
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