A pneumatically driven turbine drive system is coupled to a gas turbine engine that includes low and high pressure compressors, low and high pressure turbines, a lock-up clutch, and at least one engine accessory driven by the high pressure compressor. The pneumatically driven turbine drive system selectively bleeds air discharged from the high pressure compressor and supplies it to an air turbine that is coupled to the at least one engine accessory. Thus, the system selectively reduces the power extracted from the high pressure compressor and is capable of supplying power back to the engine core. This, coupled with the bleed air that is diverted from the high pressure turbine and the low pressure turbine, allows the high pressure spool and the low pressure spool to run at lower speeds when high engine thrust is not needed or desired, but when the at least one engine accessory is still needed.