Aaaaa, the joys of government paperwork and double speak!
North America is at least three to five years behind Europe in setting CITES into place.
The foundation (birth) document is the starting point. At the hatching of the CITES's controlled parrot, a Birth Document is provided, much the same as a Birth Certificate for a Human. That document has required information that the Breeder needs to complete as part of the document. As each new owner (buyer), including reseller, become involves, they must be listed on this document and this continues until the death of the Parrot. The register part is more specific to the Nation you live and their level of enacting CITES. At its full enactment, each owner from birth to death is recorded. Whether that is completed with local, regional or national government is likely different Country to Country.
What is important is that with in days of you obtaining your Parrot and its paperwork, you see your CAV (Certified Avian Vet) and provide them the document for them to make a copy. As I understand it, a CAV cannot provide medical care unless the Parrot has the proper documentation.
Start with the Breeder and/or Reseller, and assure that your Parrot will have a properly completed Birth Document at meets CITES requirements. Any push back in you getting that documentation should have you running for the door! Since Breeders and Resellers tend to go out of business, you become responsible if your Parrot does not have proper documents! You could be labeled a Black Marketer!