What does Parris fear about the rumor that his daughter has come under a spell?
Reverend Parris is the town's minister. This is a very important position in Puritan New England. Because he holds this high position, he is very worried about how he is seen by the people in the town.
Since he's worried about his image and about whether people will respect him, of course he doesn't want people thinking bad things about his family. This is especially true if the bad stuff is somehow related to Satan (given that Parris is the minister).
So the reason for this is that he has an image and an important place in society to uphold. Having his daughter bewitched would threaten that.