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Mostly meeting their own parents in a time travel sci-fi movie leads to awkward moments and a comedic effect in the film as used in the Back To the Future (1985), another example coming out of top of my mind is Avengers:Endgame (2019)
What was the first sci-fi story to feature a character go in the past and interact with a younger version of their parents?
This question served as inspiration for this question.
Found an early example in this chronological list of old time travel stories: "The Time Tragedy" by Raymond A. Palmer, from the December 1934 issue of Wonder Stories, available on archive.org here. A very short story where a judge in 1901 hears a noise from his father's library and finds a strange young man who has just killed the father with a "heavy andiron". The man escapes the library but is soon caught by the police, and the judge visits him in prison, but the man refuses to say much beyond his name, which is the same as the judge's name--William Gregory. A trial takes place, he is found guilty, and the judge sentences him to death.
Years later the judge has a son, also named William Gregory, who goes on to become a brilliant inventor, and in 1933 he tells his father that he has been working on the theory of time travel and shows him a machine that he intends to test. After a successful test transporting a cat on a short trip into the future, the son disappears, and the household assumes he has been abducted. But then the judge looks at an old newspaper account of the trial in 1901, and realizes that the man he ordered executed was actually his own son:
Thus, for more than a month now we have been vainly trying to solve the mystery of his "abduction" with no success until this morning, when it became necessary to retrieve those unfortunate scrapbooks from the garret in search of an item desired by the newspaper.
Having them laid before me, I took an interest in paging through them to kill the inaction of eternally waiting for news that did not come. And then the photographer delivered the proofs of the snapshot I had ordered developed. I stared long at the photo of my missing son, and then laid the photo down upon the open book beside an old newsprint photo. As my eyes compared them, the terrible realization froze my brain in my head—for the prints, though in different poses, were identical! I knew then that William Gregory, the murderer, was William Gregory, my son.
Impossible, you say? No, my dear McKennedy, I have considered it from every possible angle. There can be no mistake, though I have tried desperately to confute my reasoning. As if I had witnessed every action of my son on the day he disappeared, I know that he stepped into the time-machine determined on a trip into the past, perhaps himself choosing 1901 as his goal. Great God! Why did he not realize that the machine would no more travel with him than a cannon travels with its projectile? But he did not, and turning the switch was hurled backward in time to 1901, and through some misplacement of space during those years, was precipitated into the library just as my father entered it in search of his law book. Father, discerning an intruder, attacked immediately, actuated by his naturally impetuous nature. William, dazed by his trip and finding himself assaulted by a stranger, grasped the andiron and struck in self-defence.
Though if the question is meant to be restricted to stories where a person interacts with both parents when they were younger, this one wouldn't qualify--the time traveler doesn't meet his mother in 1901.
1944: "When the Bough Breaks", a novelette by "Lewis Padgett" (Henry Kuttner and C. L. Moore), first published in Astounding Science Fiction, November 1944, available at the Internet Archive.
Alexander Calderon, an immortal mutant homo superior, does not go back in person. His transtemporal interaction with his parents consists in sending agents back in time to his infancy to take over his parents' household and supervise his upbringing:
"The adult Alexander. The mature superman. It's a different culture, of course — beyond your comprehension. Alexander is one of the X Frees. He said to me, through the interpreting-machine, of course, 'Bordent, I wasn't recognized as a super till I was thirty years old. I had only ordinary homo sap development till then. I didn't know my potential myself. And that's bad.' It is bad, you know," Bordent digressed. "The full capabilities of an organism can't emerge unless it's given the fullest chance of expansion from birth on. Or at least from infancy. Alexander said to me, 'It's about five hundred years ago that I was born. Take a few guides and go into the past. Locate me as an infant. Give me specialized training, from the beginning. I think it'll expand me."
Well, I'll leave OP to judge whether it counts...but Robert A. Heinlein's "All You Zombies" first appeared in March 1959 in The Magazine of Fantasy and Science Fiction. In this book, the protagonist not only meets, but is his(/her) parents. From the Wikipedia summary:
Professing sympathy, the Bartender offers to take him to the abandoning seducer, whom the Unmarried Mother wishes revenge on. The Bartender guides him into a back room, where he (Bartender) uses a time machine to take them to 1963, and sets the young man loose. The Bartender goes forward eleven months, kidnaps a one-month-old baby, and takes her to 1945, leaving her at an orphanage. He returns to 1963 one month later and picks up the Unmarried Mother, who was instinctively attracted to his younger female self and has seduced and impregnated her. The Bartender nudges him to connect the dots and realize that the seducer, the young woman, the baby, and the time traveler are all him.
Not sure what level of incest we're at here, so I'll let OP judge.
I don't know if this is the first such story, but it was published in the early 1970s and the protagonist travels back in time and meets and interacts with his parents, grandfather, most of his brothers and sisters, and even his five-year-old self:
Time Enough for Love (1973) by Robert Heinlein is a novel composed of a series of stories, most of which are framed as memoirs of a long-lived man called Lazarus Long. The final story in the book is not a memoir but an adventure where Long travels back in time to the town on Earth where he grew up, and meets his family.
