References:
The Soviet Air Force Since 1918 by Alexander Boyd.

According to the above source, in early 1941, the early VDV Brigades were being formed into Airborne corps by the new Directorate for Airborne Forces (U-VDV), and they would have been organized as such:

III Airborne Corps (Kiev Military District)
       3 x Airborne Brigades

IV Airborne Corps (Belorussian Military District)
       3 x Airborne Brigades

V Airborne Corps (Leningrad Military District)
       3 x Airborne Brigades

By the outbreak of Operation BARBAROSSA in June 1941, only a single brigade in each corps had been fully equipped.