On a bitterly cold April morning in 1998, my father died of a heart attack. The shock of his death was like a punch to the stomach. It was the first bereavement I'd experienced up close. For weeks, a cloak of confusion, rage and disbelief descended. By contrast, my mother's death, five years later, held no shock. It arrived clearly signposted, with a predictability that was agonising: diagnosis, scan, operation, false hope, radiotherapy, hospice, morphine, death.

At 39 I'd become an adult orphan, a member of the club that nobody wants to join but most will. One parent dying was devastating; but when my mother died it changed me for ever. I felt anchorless, as if I was no longer anyone's child. I may have looked the same but something inside me shifted.

A friend likens being an adult orphan to being the only tree left standing in a forest. I know what she means. For me it's as if my roots have been hacked away: my parents are the reason I'm here, what held me up. They had been the one stable point during my whole life, the constant. Yes, I'm an adult and can stand alone. But there are times I still need my mother and father, times I feel very alone. I have a lovely husband and wonderful friends. I'm grateful for all of them. But they're not my parents.

My mum, Elpida, and my dad, Yiannis, came to Britain from Cyprus, separately, and met in London in the 1950s. They'd both been very poor in Cyprus, but here they had a chance to make a living. They arrived with no qualifications, no English and no money. What they did have was a strong work ethic and a lot of hope. Their lives were spent working in factories and, eventually, they were able to provide a decent home and a stable life for me and my sister, Kayti.

They weren't young when they died – in their 70s – but somehow their ageing had taken me by surprise. I remember visiting my dad one day just after he'd washed his hair and hadn't had time to slick it down with his usual squirt of Brylcreem. It was almost completely grey. When had this happened? When had he got old? The Brylcreem had always made his hair look much darker, and we used to look at old photos and joke about his "movie star" looks, while my mum rolled her eyes. To accept your parents have aged is to accept that you have too, and I suppose I've never really felt my age. But after they died I was faced with the uncomfortable reality of my own mortality. Of course, my brain knew that my parents wouldn't live for ever. My heart, however, hadn't quite caught up.

Eight years on, and it still affects me. When I hear someone whinge about visiting their parents at Christmas, it's all I can do not to groan out loud. I want to shake them (and possibly give them a good, hard slap). I want to say, "Don't you realise how lucky you are?" But, of course, I don't. Instead, I make some comment about how they should enjoy it while they can, as both of my parents have died and there's nothing I'd love more than to be in their position. An uncomfortable silence usually follows along with a muttered, "Yes, I guess you're right," and a swift change of subject.

If discussing death is still taboo in 21st-century Britain, multiply that by 10 and you get an idea of how people react when you say you've lost both parents. They just don't know what to do with that information. (You don't need to do anything, by the way – a simple "I'm sorry to hear that" is always appreciated.)

There's an awkwardness, almost embarrassment, attached to being an adult orphan – not for me, for others. I find this frustrating and stupid. In a day and age when it seems no subject is off limits for scrutiny – sex, addictions, which celeb did what to who – this most everyday of subjects is avoided. I don't wear an "adult orphan" badge. I don't go round saying, "Hello, I'm Eleni and both of my parents are dead." But if it does come up in conversation I don't shy away from it either. I believe that we're all more the same than we are different, and life stages such as this are what bring us together.

Yet I can almost taste other people's aversion if I broach the subject. As if it's bad form to talk about it at all. Maybe this is connected to the fact that we all know we'll have to confront adult orphanhood at some point. My personal experience, by the way, is that the middle-aged are the worst. People in their 40s just don't want to discuss death or bereavement, as if by talking about it, they may catch it too. Perhaps it's too close to home and they don't want to see what is waiting for them down the road. Children, on the other hand, seem more relaxed. When my eldest son saw photos of my parents he said, "Yeah, they look really old!" as if it all made sense to him. And the young will ask the two questions most of us want answers to: how old were they? What did they die of? They try to make sense of it. I've found that most people over 60 seem more relaxed to have these conversations, too, perhaps because many have been through it.

When my parents died there were some very good friends, great family members and lovely colleagues, all of whom rallied round. But there were also some hideous experiences. And unfortunately they tended to leave a more lasting impression. I remember going to work in a particular office a few weeks after my mother had died. It was a place I was known, where I'd worked shifts now and then, and where they knew what had happened as I'd worked there during my mum's illness. On my first day back, nobody said a word. Nothing. In fact, they didn't mention it the whole week. It was like that Fawlty Towers episode when John Cleese runs around yelling: "Don't mention the war!" Only one person acknowledged my bereavement, as we were buying our sandwiches one lunchtime. Adult orphans are expected to just get on with their grief quietly. We're allowed a week's grace at the most, then after that we're expected to have dealt with it. To have got over it.

To anyone who hasn't lost their parents, here's some news: you never get over it. I'm not trying to startle you. It's a fact. You get through it, yes, and you'll probably get used to it, but you don't get over it. A piece of your life jigsaw has been removed and, however much you rearrange the other pieces, they never quite fit in the same way again. That's not necessarily a bad thing. For me it makes complete sense that everything changes; if we accept that, in some profound way, our parents help shape who we are then surely their deaths will affect us deeply too?

A year after they died, my husband and I adopted our two sons, aged four and six. There's nothing quite like parental death swiftly followed by motherhood to really make you examine how you were brought up. It made me think about the values I wanted to instil in my children and what I would do differently. Despite the grief, I would say that the past eight years have been good for many reasons but especially because of the arrival of our children. So there have been many moments of joy and I think I appreciate those moments more now because I've also experienced the lows.

The difficult times are still there, but they ebb and flow and I've learned to accept them. Birthdays can be hard, as can the anniversary of a parent's death. Not every time, not every year, but occasionally. There's no rhyme or reason to when it might happen. I can be fine for months, maybe a year, then the smallest thing can make my heart dip; seeing a young child with grandparents sometimes does it because my parents never met our children. My friend, Nicole, gets tearful when she hears the Strictly Come Dancing theme tune because her mother loved the programme and they would always discuss it afterwards. Last week I was walking along the road and heard an elderly Greek man chatting loudly on his mobile phone. Most of my family lives in Cyprus, so to hear anyone speak Greek immediately takes me back to my parents. They were my link to my heritage and now they've gone, it feels as if that's fading too.

Grief can do strange things to you. An emotion that often rears its head is envy. It's not something I'm proud of, but it's there all the same. It usually burns low, but increases slightly in certain situations. My in-laws, who have always been supportive and couldn't be lovelier, are a gentle reminder of what I have lost. Family gatherings can be hard. I envy my husband his relationship with his parents and the fact that he can call them for a catch-up whenever he wants. Stupid, I know. Like a child stamping her foot, declaring, "It's not fair! I want that too!"

My parents were by no means perfect and I wasn't the ideal daughter. There had been some huge rows over the years, mostly about my unwillingness to do what was expected. ("Good" Greek girls do not leave home, buy their own flat, shack up with a boyfriend and then, when they – finally! – decide to marry him years later, refuse to do so in a Greek church.) But despite all the conflicts I think that, overall, we eventually had a good relationship. And over time, that relationship with them has continued. Because despite my initial feeling that, once they were both dead, I was no longer anyone's daughter, I now realise that isn't true. I'm still their daughter: I always will be. And they'll always be my parents. I carry them with me each day.

Cruse provides free support to anyone affected by bereavement, cruse.org.uk