I need some help understanding a sentence

As with any work of fiction, sometimes phrases do not have precise meanings which all readers would agree on. One of the interesting things about literary writing is that it has unusual combinations of words which can evoke a range of imagery. So what follows is my own interpretation - others would differ, I'm sure.

With that in mind, I would say that gaping might refer to a mouth being opened wide, or a jaw hanging low. The image is of something wide or elongated. I think the phrase gaping solemnity is intended to convey an emptiness inside a large hole, devoid of joy - Parsons (or his face) is just an empty hole being filled with the words from the announcement.

The phrase edified boredom is harder to interpret. Edified usually refers to some kind of moral teaching. It's unusual to combine this with boredom, but it might be intended to mean that the words of the announcement are like constant moral teaching which lacks anything interesting about it. He has heard these words over and over and is bored by the moral message they carry. The phrase feels like another way of saying 'indoctrination' or perhaps even 'brainwashing'.

Overall, the sentence conveys a sense that Parsons is hollow with no feeling or sentiment of his own - he is bombarded with moral teachings and his instincts have been dulled by the repetitive announcements.