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.