Unlike the earlier victims of the witch trials, John Proctor, Rebecca Nurse, and Martha Corey are all respectable citizens in Salem. Their confessions would carry enormous weight in the town, but they are desired by Hale, Parris, and Danforth for different reasons.
By the time Proctor, Nurse, and Corey are imprisoned, the trials have continued in Salem for some time, just as they have in neighboring communities, particularly in Andover. Public opinion has changed slowly; discontent, anger, and even rebellion toward the courts in these trials has grown. Word has come that the court in Andover has been overthrown.
Consequently, Judge Danforth wants the three confessions to shore up the court in Salem, thus preserving his own power, as well. Also, the confessions would affirm the rightness and justice of what has already happened there in terms of the executions carried out. They would absolve him from any error in judgment or moral responsibility for taking innocent lives.
Parris, as usual, acts out of self-interest. As discontent has grown in Salem, he has begun to feel personally threatened. When he truly feels danger to his life, he reacts by wanting to dissolve the court itself--whatever it takes to save his own skin.
Rev. Hale, the moral antithesis of both Parris and Danforth, seeks the confessions not for personal gain, but to save the accused. For Hale, even though lying is a great sin, the loss of innocent lives is a greater one. After once quitting the court in disgust, he returns to Salem, a broken and tortured man for the role he has played in the trials, and moves from cell to cell, urging the accused to confess. His desperation is shown very dramatically in his final scene with Abigail Proctor, when he begs her as a last resort to get John's confession and save his life.
In the play's conclusion, Martha, Rebecca, and John all continue to defy the court and refuse to confess. They go to their deaths, honor and integrity intact, and very soon after that, the power of the court is broken.