Has anyone got first hand experience of Napoli being that much worse than many other major cities in Italy? I found it ok, definitely better during the day than at night, bit like Florence but I think an armed military escort might not be necessary. It's not Bogota, Baghdad or Limerick (joking!)
Like ragazza I took the Eurostar and thought it was excellent, oh to have it here - 200kph+, free trolley service, reclining seats...
If you are going on to Pompeii, Herculaneum, Sorrento or the Amalfi coast the local train is crap and that train platform is pretty grotty and desolate rather than rough. There is loads of Ultras graffitti around, same as Sorrento and Herculaneum.
If you want to make your parents trip as comfortable as possible then depending on your final destination I would suggest, if direct flight unavailable, the following options
To Sorrento, Pompeii etc.
Eurostar Rome to Napoli
Ferry Napoli to Sorrento or Capri
To Positano, Ravello etc,
Eurostar Rome to Salerno
Ferry Salerno to Positano - train station is across street from ferry, easy!