If we go simply by the actual text of the story, there is absolutely no way for us to know what happened to Conradin's parents. His parents are never mentioned in any way in the story. All we know for sure is that Mrs. DeRopp (also known as "The Woman") is his cousin (we do not know if she is a first cousin or some other sort of cousin).
I do not really think that it is important to know why his parents are out of the picture. All that is really important is that they are not there and (perhaps partly because of that) Conradin is a very angry little boy.