Hän lähtee pian pesästä.
I think this is supposed to be like… parental HOME? or maybe (more metaphorically) NEST?
“Roof” doesn’t work here, because people don’t live on roofs. (Whoever wrote this sentence clearly had the expression “UNDER your parents’ roof” in mind, but inserting that expression would take some pretty complex surgery on this sentence.)
“leave the parental roof” is indeed an expression that exists. By Google hits alone, you get 8k for “leave the parental roof” and another 4k for “leaves the parental roof”.
If you substitute “roof” with “protection”, the sentence makes sense.
Looking at the search results, the expression seems to have been common at the end of the 19th century.
I’d say it’s unheard of in modern English. There are remnants of this old expression still here and there. A father might say to his disobedient children: “as long as you live under my roof, you follow my rules!”. Or, he might complain about how his children don’t appreciate him: “I put a roof over their heads, and this is the thanks I get!”
But the phrase “leave the parental roof” sounds very strange (in my opinion). I don’t think I’ve ever heard anyone say this. I agree with alewifey’s guess that “nest” would be more common in modern times: “the kids will be leaving the nest soon to go to uni. Then we’ll have some peace and quiet around here at last!”
But as I suspected, the huge vast majority of hits containing the phrase “parental roof”—definitely >>>99.9%—are preceded immediately by “under the”. Without “under” the phrase just doesn’t make any sense even metaphorically, unless you’re a microbe who lives in roof gutters.
Try searching for…
"parental roof" -under
—which will file for hits that don’t contain the word “under”—and you’ll notice that almost every hit is either a book that’s 100 or more years old, or a LiveJournal from a 15-year-old who’s still learning how to write.