If we made our travel decisions based on news reports pretty much EVERY area of the world would be off limits, including our own home towns.
So let's put it all in perspective. The overwhelming majority of murders in Mexico have to do with the drug trade and tourists are generally not involved. Yes, the fact that migrants, rival drug families, law enforcement officials, mayors and their families are being massacred is atrocious, to say the least. But what you need to look at and explain to your parents is that it does not affect your own safety to any large degree.
If the logic behind avoiding Mexico is to not be caught in the crossfire then we must also apply that same logic to other places.
British Columbia, Canada:
Vancouver: 28 drug shoutouts (give or take) in a 6 month span
Vancouver: My next door neighbour disappeared. It turns out her new husband killed her and her daughter, along with a business partner of his.
Vancouver: Innocent individual is visiting in an apartment. It turns out the renters were involved in the drug trade. Rival gang members burst in and start shooting and killing people, including the innocent visitor.
Vancouver: Over 50 women disappeared from the downtown eastide streets over several years. A serial killing pig farmer is arrested and convicted.
Victoria: 2 incidents of young women being sexually assaulted in the span of 1 week.
Victoria: Charred remains of young girl found within 2km's of my home. Two youths arrested and charged.
Interior of BC: Family of 6 kidnapped and murdered while vacationing in BC
Interior of BC: Elderly couple goes missing and their RV is ablaze. Presumed abducted and/or killed.
The list goes on and on. The point is that there is crime everywhere. Ask your parents to tell you about specific crimes committed against tourists. Then have them put those in perspective. Have them compare it to peaceful beautiful British Columbia above. It may also help to have them keep it in perspective with the crimes happening in good old Phoenix as well.
Can there be a guarantee that nothing will happen to you? of course not. But I can guarantee you that you will be putting yourself in danger by letting fear getting the best of you. It's a slow death on the inside ...I do realize that it's your parents that need convincing, not you.
In the end, it's not really any more dangerous than large US cities IMHO, as long as you use some common sense and don't become involved with drugs (they're just as illegal down there than anywhere else), idiotic foreigners who think it's OK to break the law in another country, and stay away from places and areas that give off bad vibes you'll be OK.
Tell them that this is coming from a single female who rode her bike from Victoria, BC to Yaviza Panama and back in early 2010. I experienced exactly zero dangers with people and I visited every single country down there.