Many years ago I read a story about an astronaut who comes back home after many years in space. He is very happy to meet his parents, but also worried. After the short meeting the reader finds out that the three of them are clones. I would like to know the name of the story since I can't remember it
Another possibility is "Promises to Keep", an alternative title for the short story "And Miles to Go Before I Sleep" by William F. Nolan, which was first published in Infinity Science Fiction, August 1958 and is available at the Internet Archive; but again it's robots rather than clones.
The story starts with a man going home to see his parents after many years travelling in space - as he agreed to do when he left. He is diagnosed with a fatal disease that will kill him before he gets back. So as not to disappoint his folks he arranges for a robot duplicate to be built and spends his remaining time coaching it to fool them for the duration of what would have been his leave.
It goes off OK with the town turning out to greet the returning hero and the robot leaves with his parents to go back to their house. It's then that two of the leading townspeople reveal that the parents died some time ago and also had robot duplicates built to welcome their boy back home so as not to disappoint him.
If we read "robots" for "clones", there's a pretty good match in "Look Homeward, Spaceman", a short story by Robert Silverberg; originally published (as by "Calvin Knox") in Amazing Stories, August 1956, available at the Internet Archive; reprinted in Science Fiction Greats, Winter 1969 (all-Silverberg issue), also available at the Internet Archive.
The astronaut comes home on a two-day leave after six years in space:
I guess this is the street, Paul thought. I'm back. It annoyed him that after six years in space he was no longer quite sure where his home was.
Paul comes home to his mother and his brother Jim—and, to his surprise, a robot replica of himself:
Horrorstricken, Paul stared at the man who had opened the door, and the man inside calmly returned the glance. They stood there, frozen, looking at each other.
He was looking at himself.
Not quite himself, he decided once the initial shock had worn off. It was a younger version of himself at the door, with narrower shoulders, a paler face (Paul was proud of his heavy space-tan). The livid white scar across Paul's forehead did not mark the brow of the other man. He was shorter, softer-looking.
[. . . .]
He rushed to greet his older brother, but Jim dodged around him and went to the other Paul. With a swift, sudden motion Jim touched his hand to the other Paul's back, and the other Paul stiffened in mid-stride.
"A robot!" Paul said, astonished. "A robot copy of me!"
[. . . .]
"Tell me what that thing is doing here. It was a nightmare to find myself answering the door."
Jim chuckled hollowly. "I bought him right after you left. It was pretty rough for a while, Paul: mother missed you terribly the first week. She became ill worrying about you."
"There was nothing about it in her letters," Paul said.
"Of course not. We'd never tell you. I wrote a lot of those letters myself. She was so sick that the doctors told me she wouldn't live unless you came back from space immediately. I didn't dare ask you to come back—I couldn't do that to my own brother—and so I had the robot double made. I told her that you were back, that your flight had been called off or something like that, and she accepted it."
At the end of the story, after Paul goes back into space, the reader learns that the mother is also a robot:
They went downstairs, mother and son, and Jim took out the key and activated the Paul-robot. The robot finished the word he had to say, and stretched his limbs.
"The family is complete again," Jim said.
"It's good to be moving," said the Paul-robot.
Mr. Robinson turned to her son. "You'd better renew my charge too, Jim," she said. "I'll be running down any day now."
He opened the panel in her back and touched the key to the charging-stud.
"There. Enough energy to keep you going for weeks, mother." He put his arms around his mother and his brother. "Now we can get back into our regular family routine again, until the next time Paul has a furlough. Then we can play the game again."
He looked at the Paul-robot and the mother-robot, and at the cozy four walls that bounded his little world. It was good to have the mother and brother and the home he loved, all the rest of his life. He smiled warmly, and thought of poor Paul fighting for his life on those hot, uncomfortable planets.
He drew them closer to him. "I can't tell you how much you mean to me," he said. "Both of you." His voice was heavy with emotion.