If HOA hired a pests control company to get rid of rats and a child or dog dies, who is liable?

The answer to your question depends on whether there is any scientific evidence, such as autopsy results, to show that the pesticide caused the death.

If the pest control company is negligent, it is liable, but if the HOA knew it was operating in an unsafe fashion it might also be liable. The home owners are not personally liable, but the HOA, less any insurance coverage, gets its money from the homeowners.