What is the conflict between Emily and her father in "A Rose for Emily"?
The relationship between Emily and her father should not be described as "conflicting", but rather as "codependent". The once-"mighty" Griersons were led by their patriarch, Mr. Grierson. From what we learn from the story, this was a very dominant man whose only daughter, Emily, became the epicentre of his life. A combination of Old South male-dominance combined with Emily's gender, made him a very jealous and over-protective father.
Miss Emily a slender figure in white 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.
The downside of what could have been a healthy father/daughter relationship is that, as a result of the co-dependence of allowing the father rule her life, Emily lost many opportunities to become independent, or even to marry a good man. As a result, she ages alone, eccentric, and enigmatic.
So when she got to be thirty and was still single, we were not pleased exactly, but vindicated; even with insanity in the family she wouldn't have turned down all of her chances if they had really materialized.
This means that, had Emily had better chances and more opportunities to develop, she may have turned out quite differently. Therefore, it is co-dependence, rather than conflict, what has turned Emily's life into what it now is.