The first indication Roz Chast had that her elderly parents weren't coping was when she noticed the level of grime in their apartment. Not ordinary dirt, she writes, in Can't We Talk About Something More Pleasant?, a graphic memoir of the decline and eventual death of her parents, but "a coating that happens when people haven't cleaned in a really long time. It covered everything."

In her professional life, Chast, 59, was a highly successful cartoonist for the New Yorker, at the top of her career and in control of an orderly life in leafy Connecticut. The moment she stepped into her parents' tiny apartment in Brooklyn, however – site of her unhappy childhood and where they had lived since 1959 without, apparently, throwing a single item away – she was transported back into a raging and impotent teen, "filled with dread, guilt and a weird kind of claustrophobia".

Those who know and love Chast's work think of her as the queen of family angst, a brilliant chronicler of domestic strife, and the account of her parents' last year – as they move from the apartment, to hospital, to a care home in Connecticut – is an extraordinary record of the love, fury and ambivalence that often characterises these experiences.

"Of course, there's always someone who writes in to say, 'I found it cruel,'" she says. "But mostly people are glad that I've said it was really hard, and really messy. I wanted to write about the entire experience, including the parts that were gross, and funny, and including my mixed feelings about my parents. I didn't want to write with a fake, rosy glow."

We are in the kitchen of Chast's house, overlooking her garden. She works in a sunroom, surrounded by artwork, or in an armchair in the living room, where one of her two pet parrots sometimes perches on her shoulder. (The sofa is pecked bald in places, as is the parrot, who has an anxiety disorder and plucks his own feathers, a detail so apt as to seem taken from one of Chast's own cartoons.) The biggest problem, she says, was getting her parents to face up to the fact that they were sliding into chaos. This was not a family that talked about things, certainly not about death, or money, or intimacy, except in the most general and apocalyptic terms.

George, Chast's father, was terminally anxious, while her mother, Elizabeth – "built like a fire hydrant" and with a personality to match – ruled the home with an iron will. "Her emotions were very primary colours," says Chast. "When she was angry, she was really angry. And if she felt sad, she would just be sad." She took her job as an assistant school principal extremely seriously and had no time for what she saw as the soppy self-indulgence of her husband's approach to things. "My father was the introspective one. And she would yell at him, 'George, you're walking around with your feelers out!' Well, yeah, that's what we do as people on earth. We're not all made of metal."

They were an odd pair, utterly devoted to each other and not like other parents: "My mother quoted Shakespeare and used words like 'interstices'. 'The dust will get into the interstices!' Or 'metatarsal'. Most of my friends' parents did not use that kind of language." They were also remote from Chast, not particularly nurturing, and very much parents, not friends. When she visited their apartment that time and noticed the grime, she hadn't set foot in the place for 11 years. "Aren't I the worst daughter?" (They visited her a few times a year in Connecticut.) On some instinct, however, she drove to Brooklyn that day and saw that they were in trouble. As she writes ominously in the book: "Something was coming down the pike."

George and Elizabeth Chast had saved all their lives. They didn't have a mortgage, not trusting the banks and believing that borrowing money was wrong. They made do and mended, even patching up an old oven glove to make it go the distance. They were classic products of the Depression. And so, when Chast's mother injured herself in a fall and her father started showing signs of dementia, Chast moved them to a care home near her house, where the contrast in weekly expenditure was so horrifying, she says, you could only laugh.

It was a terrible time. The Chasts had medical insurance, but, incredibly, it wouldn't transfer from New York to Connecticut. And so they were thrown into a world of almost unthinkable financial burden. "To go from a patched oven mitt and a washcloth full of soap slivers, to $14,000 a month?" says Chast. "You have to look at it as a black comedy. Scrimp and save, scrimp and save, and then it it all goes to Sunset Acres in two years, at the end of your life."

Chast did all the things one has to do; she put them somewhere decent and clean. She got them a lawyer and, despite their horror of such things, forced them to sit down and talk about their finances. As they inched into their late 90s, she arranged for round-the-clock care. And she kept a horrified eye on their dwindling savings, all the while thinking: "There goes my inheritance.

"It's a terrible, terrible thing and you look at yourself in the mirror and think: I'm a worm. I'm a lowly, shitty, crappy, horrible worm to be thinking about this. But everyone does think about it. Because you want to be able to leave some money for your kids."

What she inherited, instead, was a world of junk. As her parents lay dying, Chast dragged herself back to their apartment and started the grim task of sifting through a lifetime of worthless possessions. "You would open a drawer, which my father had jammed full of newspapers, and the bottom would drop out. There were buttons and screws and nails and bottle caps and jar lids – the drawer of jar lids! Why? Because they're made of metal and maybe there'll be another war and we'll need the metal. A friend of mine – I quote him in the book – says, 'You have found the source of the river eBay.'"

Her father died first, aged 95, and as Chast's relationship with him had been closer, she was less riven by guilt than she was during her mother's last days. The most painful part of the book is when she tries, at the 11th hour, to ask her mother if she regrets the fact that they hadn't been closer.

"That was bad. She sort of shut it down. She didn't really want to talk about it. It was too late. I think that's when I started to realise that you don't always get what you want. You don't get closure, or understanding. It just doesn't always happen."

Chast had done right by them, but she was still sick with regret after they were gone. "Should I have taken them into my house? Should I have seen them more? Why didn't I love them more? You know? Am I a disgusting person? I mean, I really loved my dad, but is there something that I'm missing here, in all this?"

The final sketches of her mother on her death bed are terribly poignant and, despite the sometimes lacerating honesty of the book, the impression one takes away from it is tenderness: the tenderness of a story truthfully told, and to that extent, a fine tribute to her parents. Their deaths made Chast look at her own two children, both in their 20s and with whom she has great relationships, and "feel so sorry for my mother. That she missed out on something that to me is the most important thing in my life."

In the wake of publication, people have written to Chast to ask for advice on how to deal with their own super-elderly parents, and she is at a loss as to what to say. That people live too long? That it is absurd to spend so much on sustaining lives it is hard to  see the value in? At the end of the day, there are no profound insights; just horror, and sadness, and grim amusement. "It was not something that I wanted to be doing," she says. "But I had to do it."

• Can't We Talk About Something More Pleasant? is published by Bloomsbury on 3 July, £18.99