The tropical climate of Central and much of South America is very conducive to plant growth so there are many beautiful gardens - particularly in Brazil, the Caribbean, and Costa Rica. There are not so many gardens of interest from a design point of view, although the Brazilian gardens of Burle Marx are a notable exception. Chile and Argentina are mostly temperate rather than tropical and have some good private gardens.
Some companies are starting to offer garden tours in South America. One advantage of booking on to a tour rather than visit as an independent traveller is the tour operators have often arranged access to private gardens that otherwise you might not be able to see or even know about in the first place.
The Amazon rainforest is the most biodiverse tropical forest in the world. South America did not suffer such climatic extremes as other tropical forest areas so there has been less extinction.