Here are two airlines I can think of that started in 1920's, not the oldest but the perhaps can be considered the 5 oldest:
Lufthansa:
Lufthansa - Chronicle
"1926: Lufthansa owes its origins to "Deutsche Luft Hansa Aktiengesellschaft" (renamed "Lufthansa" in 1933), which is formed from a merger between "Deutsche Aero Lloyd" (DAL) and "Junkers Luftverkehr" on January 6. The new airline inherits its crane logo, designed by "Deutsche Luft-Reederei" in 1919, from DAL, the blue-and-yellow house colours from Junkers."
Delta Air Lines:
Delta Stats&Facts
"1928: Delta traces its roots back to 1924, when Huff Daland Dusters was founded as the world's first aerial crop dusting organization. In 1928 the company became Delta Air Service. On June 17, 1929, Delta inaugurated airline service with the first passenger flights over a route stretching from Dallas, TX to Jackson, MS, via Shreveport and Monroe, LA. In 1941, the company moved its headquarters from Monroe to Atlanta, GA."