I think this is something all human beings go through at some time in their lives. Both about their own mortality of the mortalilty of those who are near or dear. It can take over our thoughts, it can lead us into 'what if' scenarios where we imagine tragic accidents, illnesses, and heartbreaking scenarios. I doubt there is anyone alive who hasn't had those thoughts.
I had my children fairly late, so my children's grandparents were already quite elderly when they were born. My daughter (now 19) has had regular tearful episodes - sometimes coming into my bed at night (big lump that she is 
) sobbing her heart out because she's been thinking about 'when nanny dies' etc, and also I know she has similar thoughts about me - and if anything happened to me. Yet she has lost two of her grandparents in recent years and coped with their loss - as we all do - because it is part of living.
When I get morbid thoughts - I just look around me and think to myself, everything is going to die. Every person I see, every dog, every cat, every bird, every insect. Everyone. The bloke in the flash car, the pretty girl, the teenager kicking in a football - we are all going to die and there's absolutely sweet bugger all that we can do about it, so there really isn't any point and fretting and worrying about it.
As you grow older - I'm not far off your parents' age and my mum is in her 80s and has Alzheimer's - you just get a sort of acceptance of this is how it is, and it no longer holds any fear. What is the point in worrying about something that is going to happen to all of us? We are born, we live we die and that's how it goes. No one escapes it - not the rich, not the beautiful, not the clever, no-one.
This is why we should all embrace life - grab it with both hands, follow our dreams, cherish loved ones and make the most of every second.
What upsets me is when I see people sat there engrossed in their mobile phones - wherever you go, walking down the streets, in pubs, on the train, in cafes, on the beach, sat in the park - it's like they live in some kind of half-life, half-present. There in body but their mind is off elsewhere watching a little screen, texting rubbish, taking selfies and pics of what they had for dinner and sending it to people and all that shit. It drives me nuts and I want to scream - do you think you are immortal??? Spending your life looking at a bloody screen. What a waste. Imagine being an old lady at the end of life and looking back?
Put the bloody phones and ipads and all that away people. Look around you - look at the sky, breathe in the air - you are alive!!