Interesting debate here - and q10nik is correct that "Lower East Side School Buildings" is a noun phrase which just has the subject "buildings" (the rest are adjectives/modifiers).
Does anyone have the source? This one seems a little touchy between D and E. I don't have a grammatical reason for eliminating either D or E...like many here, I'd pick D but only based on clarity. In D, the modifier starting with "on" makes a clear sentence:
The school buildings (on the lower East Side) were so crowded that...
Whereas in E the modifier placed before "school buildings" could make for a confusing read:
The lower East Side school buildings... Are these the lower buildings on the east side? The buildings on the lower east side? I suppose that too many adjectives in a row in E may cause some confusion as to which adjective describes which word, whereas the placement of "on" in choice D removes that guesswork.
That's a subtle difference and nothing that I'd say is a completely fatal flaw in E, but pitting one against the other I'd select D for that reason.