In analyzing parallels between “Old” and “New” World cultures – which are many – I used to be more of a diffusionist, suggesting influential pre-Columbian contact, but I have refined my views based on the lack of hard science in most instances, including DNA and disease markers, as well as invasive species introduction. Of course, there could have been limited post-Beringian contact, as indicated in the below documentary by the History Channel, which I found so fascinating that I bought it years ago.
That being said, parallels in religion, myth, art and architecture are generally homegrown, based on the human mind and natural environment, not on cultural diffusionism. There simply is no good and scientific reason to assume that Old Worlders influenced significantly the cultures of the Maya, Inca or Aztecs. To do so without hard evidence is to deride the accomplishments of native peoples in Central and South America.
Further Reading
Parallels between Old and New World Religions
Astrotheology of the Ancients
Pre-Columbian trans-oceanic contact theories
Settlement of the Americas
Olmec Colossal ‘Black’ Heads Mystery Solved?
Global Comparative Religion and Mythology