Reverend Parris wants to delay the executions because he fears rebellion, and -- probably, just as he feared the loss of his own credibility and authority at the beginning of the play, he likely fears the same thing again.  He is concerned because Abigail, the chief witness and accuser, "has vanished."  She's robbed him of his life's savings and disappeared along with her best friend, and this weakens her credibility, and in doing so, weakens the credibility of every accusation and conviction that has taken place thus far.  

Further, Parris also wants to delay the hangings because there has been "rebellion in Andover," a nearby town also conducting trials.  Parris tells Danforth that "There be a faction here, feeding on that news, and I tell you true, sir, I fear there will be riot here."  Parris knows that a rebellion would likely unseat not just the court but himself as well.

Parris also worries because "it were another sort that hanged till now."  Those who have been executed were looked down upon, alienated, or outcasts of the town already.   Those scheduled to die today, including John Proctor (though not his wife, as she is pregnant) and Rebecca Nurse, are well-respected members of the community.  Moreover, the turnout for John's excommunication from the church was very small, "hardly thirty people," indicating a high level of community discomfort as well.

Finally, Parris wants to delay executions because he fears for his own life.  He says, "Tonight, when I open my door to leave my house -- a dagger clattered to the ground [...].  You cannot hang this sort.  There is danger for me.  I dare not step outside at night!"  He may have treated others' lives as though they were of no importance, but Parris is protective of his own, and he is threatened.

In Act IV, we see a very different Salem from the beginning of the play.  As the act begins, we learn that on this day John Proctor, Elizabeth Proctor, Rebecca Nurse, and Martha Corey are to be executed.  Reverend Parris, who in the previous act was quick to point the finger and condemn the accused as witches has changed his plea.  Now, after finding a note and dagger on his front door, he believes that the courts would be wise to wait to execute these even if just for a little while. Parris realizes that the people are no longer with him, and this causes to fear for his safety.  If the courts could get someone to confess, people may believe there are witches again.  Even better, if the courts can get John Proctor to confess, people will be even more inclined to believe in the cause again. He tells the court-

It is a great service, sir—it is a weighty name, it will strike the village that he confess. I beg you, let him sign it. The sun is up, Excellency!