Danforth is interested in maintaining the authority of the court. To admit that innocent people have been sentenced and killed as the result of fraud would greatly undermine the authority of the court in the public's esteem.
As the court is generally reliant upon public opinion as the basis for its authority, Danforth energetically defends the court's position that it has not done anything falsely and that the evidence used in the court has all been valid.
Parris wants to continue to side with the court and associate himself with this body of power. He fears that, as something of an outsider, his position in Salem is insecure. By aligning himself with the power of the court, he bolsters his own position as a minor authority in the town. For him, undoing the work of the court and recognizing the trials as fraudulent will reverse the progress he has made regarding the stability of his position. His position would become even more unstable than it was at the opening of the play.
Hathorne's reasons for disbelieving the claims made by Proctor and Mary Warren are more legally oriented. He simply does not believe Mary Warren. She either is lying in her confession in Act III, or she was lying in court earlier. Both ways she is untrustworthy and, for Hathorne, unconvincing.
First you have to understand the context of the entire play “the Crucible”. The groups of people that inhabit that town come from a tradition of Puritanism, so witchcraft was a big ‘no-no’ in that context. But that also implies that the members of the legal system Hathorne, Parris and Danforth are in place to uphold and maintain strict anti-witchcraft mentality and in turn, maintain the image of their religion. So, despite the fact that Abigail DID lie, what Hathorne, and Danforth are worried about the most is maintaining this image... even is if they have to execute everyone in town to do it (which they do). If they admit that Abigail is lying, and that everyone they executed died needlessly, then the entire system falls apart.