This is actually something that the text remains silent about. We are never definitively told the answer to this question. All we know is that Clark, the narrator of this story about the enduring beauty of music and the sacrifice of Aunt Georgiana by going and living West and homesteading for so many years, is that he spent a number of years with his Aunt and Uncle in their home in Nebraska as a child. Consider the following reminiscence that Clark has of Aunt Georgiana:
I owed to this woman most of the good that ever came my way in my boyhood, and had a reverential affection for her. During the years when I was riding herd for my uncle, my aunt, after cooking the three meals--the first of which was ready at six o'clock in the morning-and putting the six children to bed, would often stand until midnight at her ironing board, with me at the kitchen table beside her, hearing me recite Latin declensions and conjugations, gently shaking me when my drowsy head sank down over a page of irregular verbs.
Clark recalls a period of time where he worked on the farm with his uncle and helped him out. We can therefore infer that perhaps he either needed to stay with them whilst his parents went elsewhere and worked, or that he is indeed an orphan and he was brought up with his aunt and uncle, which is why Georgiana was such an important figure in his life. The text remains silent about what happened to Clark's parents, but it does clearly indicate that his Aunt Georgiana was an incredibly influential and formative individual in his life.