While I'd appreciate being notified if a parent has a heart attack, there's a whole bunch of more important people that need to be called before me - get them an ambulance before you call me - and, truth is, there's very little I could do in terms of "emergency response" anyway.
I can't off-hand think of anything where contacting me personally would be required in an "emergency situation", if it's really an "emergency" there's probably better qualified people to call. This is colored by my situation, I'm not a parent, I'm not the sole decision maker for anything mission critical (either personally or professionally). I'd be sad (or sadder) if my Mum or Dad died and I missed out on saying goodbye because nobody could contact me immediately, but they live 5hrs away from me so that's a pretty unlikely scenario.
I'm kinda "at peace" with spending time "uncontactable" anyway, I spend at least an hour a day (and sometimes 8 or 10 hours) on a motorcycle - where I might feel my phone ringing in my pocket, but I don't answer it until I've got to where I'm going and stopped the bike. I also treat the phone the same way in movies/restaurants/gigs/meetings, and often while deep in something complicated at work.
Personally, I don't buy into the idea that my phone is so other people can reliably contact me immediately in case of emergency. When work wants that, it's called "on-call" and they pay me for it. In my personal life, I answer my phone if and when it suits me, and I'm not enslaved to anybody who happens to have my number.