Why are Danforth and Parris so desperate to have Elizabeth speak to John and perhaps earn his confession?

Danforth and Parris believe that if John Proctor will confess to witchcraft, it will lend legitimacy to the trials, which are becoming increasingly unpopular with the citizens of Salem. Fewer and fewer people are inclined to believe that upright citizens, especially Martha Corey, Rebecca Nurse, and Elizabeth Proctor are actually guilty. When Proctor asks why the court will not consider possible ulterior motives of the accusers, the public's faith in the fairness of the trials really begins to slip. If John says that he is a witch, Parris and Danforth believe that the citizens will entertain the thought that other prominent people are, too.
Danforth and Parris are also afraid that riots will soon break out, and Parris knows that his life has been threatened when he finds a dagger stuck into his door. Because Elizabeth is pregnant and the Proctors have other children, Parris and Danforth believe that she can persuade John to confess so that he will live, but likely suffer banishment from the colony. Parris and Danforth can save face, prop up the theocracy of Salem, preserve themselves and their positions, and excommunicate an enemy if Elizabeth will persuade John to confess.