Why is Reverend Parris praying and weeping as "The Crucible" begins?
Parris' angst is reserved solely for himself, although, ostensibly, he is concerned for his daughter Betty, who is rigid under some sort of paralysis.
Parris does not want to believe, as Susanna, a neighborhood girl, claims, that he might want to "look to unnatural things for the cause of it."
Despite Parris' frantic statements to the contrary, the community believes that there is indeed cause to think that the "unnatural" (ie, the devil) is involved in Betty's illness. And Parris himself knows this might be the case, for he knows that both his daughter, and his niece, Abigail, had been discovered dancing in a pagan rite in the forest.
But, again, it is not really for Betty that Parris' cries. It is for himself, and his lauded position of leadership within the Puritan community. If he allowed such a thing to happen, and within his own family no less, he must personally be tainted by sin.
To understand this concept, one must know something about the belief system of the Puritan religion. Briefly, the Puritans believed that there were only so many "spots" in Heaven, but no one knew who had those coveted places in Eternity. Those who lived the "purest" lives on Earth were thought to be good contenders.
So, Parris' indeed has a lot to be despondent about: not only would he lose his earthly position, but his heavenly one as well.