What parallels might be drawn between Deacon Peabody and the tree bearing his name?
The tree bearing Deacon Peabody's name is quite blatantly representative of Peabody himself; the tree is like his person, and the condition of the tree reflects the qualities in him.
Peabody is said to be damned (sent to Hell for his sins), or he will be, if he continues to be more concerned with the sins of others rather than his own sins. This is represented by the rot that has taken hold of the tree; like Peabody, it is corrupted on the inside, but has a pleasant exterior. Likewise, Peabody probably uses his position as a deacon (a person of authority in a church) to present himself as a person of good moral character.
The tree is also said to be nearly cut all the way through, so that it might fall over soon; because another fallen tree is said to be representative of a person that had recently died, this might suggest that Peabody is near death as well, even if he doesn't realize it, and that there may not be enough time for him to recover from his internal rot before he dies in that condition.
Further, Old Scratch mentions that they are "ready for burning", which on the surface suggests that the trees are being used as firewood, but it really means that their souls are ready for burning in Hell.