It is a bit curious, and hard to parse at first.
The single phrase, "The son of itinerant actors," gives rise to two separate appositives, one following the other.
The second appositive matches up with the first noun, "the son." (The son, Edgar Allan Poe).
The first appositive matches up with the second noun, "actors." (Actors, David & Elizabeth).
It would probably help to get rid of the first comma, at least making the first match clear. But better yet, find another way to say it.
The rest of your excerpt is fine by me. 
Edit. Perhaps you could use a nested appositive, if I may coin a phrase.
Edgar Allan Poe, the son of itinerant actors David Poe Jr. and Elizabeth Hopkins Poe, was abandoned etc.
Though still cumbersome because of the long names of the parents, I think it flows a little better than the first. At least the appositives directly follow their antecedents.
(Well, it isn't really a nested appositive.) 