Your interpretation of this sentence's meaning is close, but I think that it's a bit off.
The nominal clause "how toward the end of his parents’ lives his mother and father spoke only through the dog" functions as the direct object of "told". It is introduced by the subordinating conjunction "how". It includes a subject ("his mother and father"), a simple predicate ("spoke"), and two adverbial prepositional phrases ("toward the end of his parents’ lives" and "only through the dog").
"Daniel a late addition" is an absolute nominal phrase. It is headed by "Daniel", which takes the appositive "a late addition".
I think that DjinTonic is suggesting that the prepositional phrase "toward the end of his parents’ lives" be surrounded by paired dashes or commas; this is often done to make such a phrase nonrestrictive or parenthetical:
Around the time they first met, Daniel had told Mary Jane how--toward the end of his parents’ lives--Daniel a late addition--his mother and father spoke only through the dog.
Around the time they first met, Daniel had told Mary Jane how, toward the end of his parents’ lives--Daniel a late addition--his mother and father spoke only through the dog.
Note that in either case, the pair's second mark gets absorbed into the dash after "lives".[1]
(By the way, I do not like the verb tenses in this sentence, but I'll keep quiet about that since you didn't ask about it.)
This is just one interpretation; I'm sure that others will analyze this sentence differently and use some different terminology.
[1] This is a bit of a simplification; I would actually consider "Daniel a late addition" to be a dependent of the preceding prepositional phrase, so that the new paired punctuation would really surround both.