This question refers to Act IV of this play, when Parris is well aware that the popularity of the hangings and of the witch hunt that he is at the centre of is waning amongst the people, and he is very concerned about the impact that the deaths of well respected people such as John Proctor and Goody Nurse will have on the public. Interestingly, he clearly states to Danforth that he wants a confession, not in the interests of truth, but because he is afraid of what the public might think:
Now Mr. Hale's returned, there is hope, I think--for if he bring even one of these to God, that confession surely damns the others in the public eye, and none may doubt more that they are all linked to Hell. This way, unconfessed and claiming innocence, doubts are multiplied, many honest people will weep for them, and our good purpose is lost in their tears.
Parris clearly states therefore that he wants one of the three people due to be hanged on this day to confess as it will send a very clear message to the public that the accusations against them are true. If one does confess that therefore strongly suggests the others are guilty too. However, if they all die unrepentant without confessing to the crimes they didn't commit, the public will doubt more and more whether the hangings are just and whether the "crimes" of such individuals are actually based in reality. This would greatly threaten the position of people such as Parris.