Mites and Fleas
Chinchillas do not carry mites or fleas, but I have seen a couple of cases where a bag of shavings or other pets have introduced them into a household. They can also come from abandoned bird's nests as the few I have seen living on baby chinchillas have been identified as bird mites.
I was asked in 2003 to assist with a mite infestation at a house that had rats, chinchillas and cats. The cats, rats and humans were getting bald spots and sores, whereas the chinchillas showed no adverse effects, yet the babies definitely had mites on them. When the vet ran out of options we discovered that the mites were killed by a chemical in a common over-the-counter bug spray called Permethrin.
All of the animals were removed for two days and the house was bombed. They did not experience any further issues.