Here in post #2 the second link is to a PDF that outlines the requirements for importing dogs, cats and other species into Ecuador, see under "OTRAS ESPECIES ANIMALES" for birds. When I click on the link, the PDF is downloaded automatically - I don't know if in your browser it may just show the PDF and then you can choose to download:
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The PDF (importacion-exportacion-mascotas-24.02.2012) is from the Ministerio de Agricultura, Ganaderia, Acuacultura y Pesca and has the requirements and their contact information. You also have to get import permission from the Ministerio de Ambiente. This is because Ecuador considers only dogs and cats to be pets (From the PDF: "Ya que en Ecuador únicamente son consideradas mascotas perros y gatos...").Â
To get import permission, inside the PDF is a link ("DESCARGAR") to another PDF form to be filled out (SOLICITUID ESPECIES CITES COMO MASCOTAS 24.02.2014).
Even if the animal being imported is not on the CITES list of protected species, it appears you are still required to fill out the form but will not need any export permission from the appropriate governmental authority in the country of export - referred to as the "Autoridad Ambiental del paÃs exportador" in the PDF. If you search the internet for "especies CITES" you will get a hit in English as the first in the list, along with others in Spanish that tell you what species of animals are on the CITES list.
To see the CITES Appendices with the species lists, go here:
You can then search within the web page for example, for Psittaciformes to see the scientific names of the cockatoos, macaws, parakeets and parrots, etc. that are on the CITES list.
In any case I would suggest contacting both the Ministerio de Agricultura, Ganaderia, Acuacultura y Pesca and the Ministerio de Ambiente both of whose contact information is on the PDFs. You should be able to discuss the matter with them to ensure you take all the steps necessary and have everyone sign off on importing the birds as pets.
It sounds like it can be done, but obviously you will need to be able to get all the necessary forms from the USA bureaucracy and then navigate the Ecuadorian bureaucracy in Spanish with persistence and patience. This looks like a difficult process even for someone who speaks fluent Spanish - but the steps are all specified in the documents for the importation, so it can be done (probably).