What are Hale's and Parris' personal reasons for wanting to obtain a confession in The Crucible?
I think that you are referring to the end of the play, when both men are encouraging John Proctor to confess to witchcraft and sign his confession. I believe that Hale now genuinely believes that he was wrong about witchcraft in Salem.
Hale: It is a lie! They are innocent!
He was called there as an expert at identifying whether or not witches were actually in Salem. He was sold on the fact that they were, but after witnessing first hand the good, Christian character of many of Salem's accused, Hale no longer believes the legitimacy of the accusations. He also no longer believes that Abigail is a reliable "witness." That suspicion is confirmed when Abigail runs away. Hale wants John to confess so that he can live.
Life, woman, life is God's most precious gift; no principle, however glorious, may justify the taking of it. I beg you, woman, prevail upon your husband to confess. Let him give his lie.
Hale also believes that his confession will encourage the other accused to do the same thing as well. Hale simply wants to see these people live.
Parris also wants to see them live, but I believe his reasons are slightly more selfish. Parris has been worried about his reputation from the very first lines of the play.
Parris: There is a faction that is sworn to drive me from my pulpit. Do you understand that?
He first wants to be known as the minister who helped purge Salem of evil. I believe that by the end of the play, Parris is more worried about going down in history as the minister who had half of his congregation killed. He absolutely wants John to confess in order to live, but it's not pure altruism.
Parris, feverishly: It is a great service, sir. It is a weighty name; it will strike the village that Proctor confess. I beg you, let him sign it. The sun is up, Excellency!