Things to know about your pet chinchilla
Can chins get * Mites * Fleas * Worms ?
 
One thing that is often said about chinchillas is the hair is too thick and heavy to get bugs in the fur.  While this is true, that chins will not get the same kind of fleas as dogs and cats, they can get skin mites.
Regardless of what you have heard, I am here to tell you that I have seen them first hand.  I was at my chin vets office, and saw him remove blood sucking mites from the fur coat of a chin while he wore special magnifying goggles, and took me into his work area, and showed me under a microscope the mites themselves. 
There are several types of these mites, and they can come from many sources, including other rodent type animals, feed, hay, pellets, bedding and from grass and soil.
The process to rid your animals of these mites is rigorous and long. It includes treatment with ivomectin oral on the animals, thorough repeated cleanings of all spaces and equipment involving the animals as well as treating the animals with each dust bath by adding a special insecticide powder.

Treatment for intestinal parisites known as worms also would take the attention of a qualified vet.