What can you infer about Roger's parents?

It can be inferred from the text that Roger is not well cared for by his parents.
The first example of this appears in the second sentence where it is stated that it, “…was about eleven o’clock at night….” We are told later in the story that Roger appears to be fourteen or fifteen years old. The fact that a fourteen or fifteen-year-old child is in the streets and unsupervised at that hour makes it reasonable to infer that his parents are not ensuring that he is safe and not roaming the streets. 
In addition, Mrs. Jones tells Roger that his face is dirty and states “Ain’t you got nobody home to tell you to wash your face?” to which Roger responds “No’m.” The first fact provided, that Roger’s face is dirty, allows the reader to infer that he does not have anybody at home to take care of him by making sure that he is clean. Also, Roger’s response to Mrs. Jones’ question directly states that he does not have anyone at home.
Furthermore, the text describes Roger as “…frail and willow-wild…” The word frail is used to describe a person who is weak while the term willow-wild describes a person who is thin. Therefore, the reader is told that Roger appears to be weak and thin. This inference is supported later in the text by Roger’s statement that there is, “…nobody home at my house…” when asked by Mrs. Jones if he had eaten supper. Considering that Roger is fourteen or fifteen years old, the description of him as weak and thing serves as a representation that Roger is generally not well fed or cared for. While it may be expected for an elderly individual to appear weak and thin, this is not something we expect from a healthy teenager.
Finally, the inference that Roger’s parents are largely absent from his life is supported by the fact that he responds positively to Mrs. Jones’ interactions with him. Not only does Roger follow Mrs. Jones’ directions to wash his face and eat, but he also decides not to run away when he is given the opportunity. Additionally, we are told that Roger does not want to be mistrusted and that he thanks Mrs. Jones before leaving later that night. This behavior allows the reader to infer that Roger is not accustomed to being cared for in the way that he has been cared for by Mrs. Jones and that he wants to return her kindness by being trustworthy and showing her gratitude.

Though Roger's parents are absent on the day he attempts to snatch Mrs. Luella Bates Washington Jones's purse, it is evident that they have tried to instill manners in their son.  When Mrs. Jones asks him questions, he answers them "Yes'm" and "No'm" (as a diminutive of ma'am) as a sign of respect for an older woman.  Once they have established some trust, he asks her if she needs him to go to the store for her. He does not want her to mistrust him, and he seems sincere. 
Roger's response to Mrs. Jones's insistence that he wash his face and eat some dinner is to comply.  He tells her that "there’s nobody home at my house."  Readers can infer that his parents are probably working and that he is on his own.  
After Mrs. Jones has given him ten dollars to buy the shoes that he wants, he thanks her.  His gratitude for her kindness is further evidence that though Roger's family is likely financially disadvantaged, they have raised him to behave civilly. They would likely be ashamed of his youthful mistake in trying to take Mrs. Jones's pocketbook.