Well, I would not let my daughter move from NYC to Mexico 
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But, maybe the article below helps to establish context for your parents.
While the best-known cities will naturally be in the news the most, and the largest cities automatically will have the largest number of incidents, one also has to consider that (except for the State of México and México City) NYC has a larger population than virtually any Mexican state – or the 7 smallest Mexican states combined. In that context, the number of incidents in NYC might compare quite favorably?
Generally NYC is considered as “safe” as most places - in fact, it is often ranked as one of the safer cities in the U.S., having fewer incidents (per capita) than many much-smaller cities that from the (far) distance, someone might think of as almost “idyllic”: