Miss Emily Grierson's father is only ever referred to in the text as "her father" or "Miss Emily's father," though he is described as the other citizens of town imagine him. The narrator says that those in town thought of Miss Emily and her father as a "tableau, Miss Emily a slender figure in the background, her father a spraddled silhouette in the foreground, his back to her and clutching a horsewhip, the two of them framed by the back-flung front door." In other words, they think of Miss Emily's father as a type, a sort of legendary and imperious Confederate era figure; perhaps this is why he is never named. What he represents both in the community and in Miss Emily's life are more important than who he actually was as an individual person. His personal identity is relatively insignificant. His name, therefore, does not matter; what matters is that "None of the young men were quite good enough for Miss Emily" in his eyes and he had "driven [them all] away," leaving her totally alone. His pride of place overwhelmed most everything else.

Miss Emily was Miss Emily Grierson.  Her name is provided in the first line of the story.  Therefore, we can conclude that her father was Mr. Grierson. That is all the information that the story gives us.