In DuckTales, Huey, Dewey, and Louie were the nephews of Uncle Scrooge. What happened to their parents?
According to the Disney Wiki, there's no information about what happened to them other than their father was in hospital once, and the information that does exist is contradictory (emphasis mine):
Huey, Dewey and Louie are the sons of Donald's sister Della Duck; in Donald's Nephews, their mother is instead named Dumbella. In the original theatrical shorts, they were originally sent to visit Donald for only one day; in the comics, the three were sent to stay with Donald on a temporary basis until their father came back from the hospital (the boys ended up sending him there after a practical joke of putting firecrackers under his chair). In both the comics and animated shorts, the boys' parents were never heard from or referred to again after these instances, resulting in the boys ending up permanently living with Donald, in keeping with Disney's usual elimination of characters' parents.
Though there is a mention in a Dutch comic (it's unclear if it's canon) called 80 is prachtig in which their mother becomes an astronaut and leaves them with Donald while she goes on holiday in space. With my translate-fu, the comic is something like:
Donald: I did not know you had kids! Do you mean my NEPHEWS?
Della: We'll talk some other time, Donald, when I come back from my spaceflight!
Donald: What are you saying? Are you going to SPACE?
Della: I miss my cuties already!
Della: So far all good! I'm on holiday until the tank is half empty! Then I'll return to Earth!
If there's a Dutch speaker out there who can help me get it closer I'd appreciate it
The pilot of DuckTales, The Treasure of the Golden Suns, has Donald joining the Navy and the boys going to stay with Scrooge while he's away.
Its a plot point in the new Ducktales!
Meanwhile, Dewey goes to Webby Vanderquack to see if she has any information about his mother Della Duck. Dewey mentions that Donald never talks about her and that all Dewey has known about her so far is a picture of Della pushing Donald into a cake. Webby is fascinated by this evidence; sadly, she has no information about Della in her "Della Duck" files, as virtually any record of Della is nonexistent, and traces have been largely covered up by Scrooge
An expansion on another answer since I cannot comment: We now know that they revived the space explanation for Della’s disappearance in the new Ducktales series. In current canon, she was lost in space during a cosmic storm while flying a rocket Scrooge McDuck built for her called the Spear of Selene shortly before Huey, Dewie, and Louie were hatched. Scrooge nearly bankrupted himself searching for her with no success, losing the rest of his family (Donald and the nephews) in the process. That is until the beginning of DuckTales when Donald had no choice but to re-establish contact after 10 years.
Nevertheless, we still have no new details about the father of Hubert, Dewford, and Llewelyn Duck since the beginning of Ducktales. Given Disney’s tendency to kill the parents of animated characters as a main plot point, it’s unlikely their father is deceased. Except for the fact that some canon does exist (although it would seem and I would claim it’s no longer canon given Ducktales’ developments), I think the easiest (and appropriately modern) solution would be to make the relationship between their father and Della one of a more temporary nature. This would also be consistent with an adventuring “no one ties me down” spirit Della seems to have. Perhaps the revelation in the final episode of season 1 of Della’s soon-to-be return will finally lead to an answer for this age-old question.
It has been suggested (see Wikipedia) that the father of the triplets is brother to Daisy Duck, who has not appeared in the Ducktales series.