Following the same procedure as the longitude question and the same elimination of nonsense and duplicates:
SELECT ST_Y(source_geom) as src_y, ST_Y(destination_geom) as dst_y, abs(ST_Y(source_geom)-ST_Y(destination_geom)) as delta, name, destination_airport, source_airport from flights order by delta limit 25;
Our first result, between Kasos Island Public Airport and Karpathos Island National Airport, two Greek islands, is shockingly, incredibly straight, but the airports are only 22km apart, failing the distance test.
Excluding that gives us a new candidate: flights between Bilbao Airport and A Coruña Airport in northern Spain, a distance of 444km and a difference in latitude of just .001. However, nobody seems to be operating that route at present. FlightAware and FlightRadar24 list no such flights, and the OneWorld route map shows only an Iberia flight from A Coruña to Madrid.
Next is PPW-WRY in Scotland, home to the shortest scheduled passenger flight in the world. Certainly fun, but less than 3km.
Which brings us to our actual winner: Dallas/Fort Worth-Charleston, United States. This provides an impressive distance of 1,589km between the two US cities with a latitude difference of 0.0018. The route is served by twice-daily nonstops on American Airlines, AAL2762/AAL2763/AAL2764 (for whatever reason, 2764 is used for both directions).
Update: Going by azimuth, PostGIS tells me the winner is TAS-UGC, a bearing of 90.007, assuming WGS84 geography.