The remnants of the maya and the inca civilizations were both conquered by the

Spanish conquistadors

Explanation:

The Maya civilization was not its peak, but it was a civilization on the demise when the Spanish came, while the Inca civilization was at its golden days. The Spanish conquistadors didn't really cared about these civilizations and their advancements, instead they only wanted their wealth, territory, and labor force. The Spanish were merciless toward the native people, and both the Maya and the Inca suffered great losses, as well as destruction of their empires, and neglect of their culture. The Spanish imposed their own culture, religion, language, political system over them and forced them to assimilate, making big damage on their cultural heritage that was built for thousands of years.