A Theory: Father knows what it is. (Spoilers)
So, I think the Father knows what the Creeper is, taking into account his age (I’m assuming he is in his mid-fifties when this story takes place).
Say, for instance, Father was 9 years old when some murmurings of disappearances and strange phenomena spread throughout the county, maybe even a rumor of a winged beast, but those memories dissolved into the realm of fairy-tale for his child mind.
Then when he’s 32, perhaps he paid a slight bit more attention when he heard of a prom couple who went missing and they never found the girl’s head, or whatever the story was, along with other odd stories of disappearances and people who claimed to have seen a man sniffing people’s belongings.
But he still shrugs it all off as urban legends, until one day, now aged 55, he actually witnesses the Creeper fly off with his child; he knows what it is, he is angry that he didn’t believe, and is determined to get revenge—not out of madness or shock, but because he is remembering all of those murmurings and fairy-tales from his adolescence, and he believes now that it is his responsibility to at least try to kill the thing, or die fighting.
. . .
This is just how I like to view the film now anytime I rewatch it—this theory isn’t really supported directly by anything in the film, it’s just my own made-up backstory for the father character, but I think it’s plausible. There seems to be more than just shock and disbelief when his son is stolen; I think there is a tinge of forgotten knowledge that turns into lament, and then ultimately rage.