Homer's rotting corpse caused the smell, not Emily's father. It was explained earlier in the story that the body of Emily's father had to be removed by authorities when she refused to cooperate and release it voluntarily; her father ostensibly received a proper burial. Emily's reluctance to give up her father's body foreshadows her later act of desperate quasi-necrophilia in order to hold on to the one person who had shown her affection.
Emily eventually used the arsenic not for killing rats but for poisoning Homer after he apparently had spurned her desire for marriage. It is explained that the smell occurred at some point after Homer's departure. Unlike her father, Homer remained with Emily long after his life had ended. His final resting place became her bed, and though she was able to hide the body for several decades, she could not mask the smell which permeated both the house and the outside property as well.
In the short story "A Rose for Emily," the reader is not told what the smell is that is coming from Miss Emily's home. It cannot, however, be her father. Emily's dad passed away two years before this smell occurred and although she would not at first let anyone take her father's body, she did eventually relent to having him buried.
The reader can however infer that the smell is from the rotting body of her male friend, Homer. Since he has disappeared, has not been seen, and shortly thereafter a rotten smell comes from her house, we are pretty safe in saying it is the decaying body of Homer. She wouldn't have told anyone because she was not about to have him taken from her.
The smell was definitely that of Homer. Emily's father had been dead and buried for two years before the smell appeared. Even though Emily initially refused to acknowledge her father's death and allow his burial, "Just as they were about to resort to law and force, she broke down and they buried her father quickly." The smell did not appear until "a short time after her sweetheart. . . had deserted her." That was after she had bought the rat poison and Homer had disappeared.
It was Homer