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.