Parson Hooper's black veil, which covers his face down to his mouth, is a symbol of his own sin and, by extension, of everyone's sins, because everyone has something about themselves they want to hide. In general, the black veil is creepy enough that it makes people uncomfortable, to the extent that they don't want to be around him. However, sinners who are suffering remorse for what they have done are drawn to him as a kindred spirit, as are dying "sinners":
By the aid of his mysterious emblem—for there was no other apparent cause—he became a man of awful power over souls that were in agony for sin. His converts always regarded him with a dread peculiar to themselves, affirming ... that, before he brought them to celestial light, they had been with him behind the black veil .... Dying sinners cried aloud for Mr. Hooper, and would not yield their breath till he appeared.
Those who live in denial and think they are good people try to avoid Parson Hooper, because they are working hard to avoid dealing with their own misdeeds or inner darkness. It is notable, too, that not all the dying call out for Mr. Hooper, but only the dying "sinners." While the narrator does not spell out precisely why the sinners call for him, we can surmise it is because Mr. Hooper acknowledges his own sin through the veil. The dying sinners can feel safe that the parson will hear them without judgment and that he understands the spiritual pain they are undergoing. His veil, if terrifying, is a reminder that they are not alone as sinners, which perhaps, too, makes it easier for them to believe their own sins will be forgiven.