At the beginning of the play, Abigail's position in the Parris household is given exposition along with Parris position in the town. Each of them occupies a precarious place.
Abigail is already suffering from a bad reputation. Parris points this out to Abigail in their first conversation, speaking of the rumors he has heard. Elizabeth Proctor is refusing to come to church because she "will not sit so close to something soiled," referring to Abigail.
Abigail's reputation affects the current situation. People will believe that something as troubling as witchcraft might have actually taken place because Abigail is such a dubious persona in Salem.
He knows not only that witchcraft is punishable by death, but also that the consequences of such news getting out in the town are dire to his own reputation.
Parris does not have a strong footing either. He worries that if the accusations of witchcraft are made against his daughter, everyone will believe them. He will be forced to give up his position in Salem and will be ruined.
... interrogating Abigail about her dealings with witches in the opening scene, he seems to worry more about what these activities will mean to his reputation than Abigail's spiritual state.
Significantly, we see that Parris suspects the truth that witchcraft was taking place in the woods that night and that his daughter, Abigail and Tituba were all a part of an occult ceremony. He does not believe Abigail's denials.
He presses Abigail for the truth because, in part, he is worried about losing his position and, also, because he intuitively knows that Abigail is lying.
Given the opportunity to use her lies to his own advantage, however, Parris drops his pursuit of the truth in favor of the safety of Abigail's lies and accusations against others.