Why is Parris reluctant to tell anyone that Betty may be bewitched?
The Reverend Parris is paranoid about the fact that there are people in the community who want to get rid of him. He will, therefore, do everything in his power to ensure that he keeps a clean record lest they use whatever they may find to be inappropriate against him. As Arthur Miller explains in the introduction:
In history he cut a villainous path, and there is very little good to be said for him. He believed he was being persecuted wherever he went, despite his best efforts to win people and God to his side.
It is this obsession which makes the reverend believe that there is a faction out to get him. If they should learn that his daughter had been bewitched, they might start questioning the strength of his faith. Furthermore, the question about how she was bewitched could lead to the discovery that she had actually been involved in improper acts in the woods involving his servant from Barbados, Tituba, and his niece, Abigail Williams. The knowledge that the largest portion of his household had been involved in inappropriate deeds, such as dancing around a fire in the middle of the night and attempting to call up spirits, would surely damn him. Also, the fact that he was the one who discovered them performing such unfitting acts and didn't mention them would lead to greater criticism and condemnation—an opportunity he believes his enemies would relish with glee.
In order to protect his position and his integrity, the reverend therefore makes it a point that Abigail not utter a word about their shenanigans in the woods. He tells her:
Now look you, child, your punishment will come in its time. But if you trafficked with spirits in the forest I must know it now, for surely my enemies will, and they will ruin me with it.
It is imperative that no one knows what exactly happened and the reverend is therefore desperate that first the doctor, and later Reverend Hale of Beverley, who he has summoned, find an answer to Betty's condition, and it must not be related to witchcraft whatsoever.
There are several reasons. First, as a minister, how would it look to his congregation if his daughter is bewitched? Second, his standing in the community is tenuous. Some, like John Proctor, have stopped going to church services because of their dislike for him. He doesn't want to give others more reasons to stop attending. Remember too that he was not the first choice for minister, Mr. Putnam wanted someone else, so he does not have the most support from the Salem community. He is also the third minister (I believe) in a rather short period of time. So again he is not overly anxious to give the townspeople reason to turn against him. Third, he is worried that people will start a witch hunt and bring in Mr. Hale, thereby taking away some of his authority.