B737-700 Captain reported a #1 Engine Flame Out during gate arrival. The #2 Engine was shut down and the APU was on line. As the flight crew was being marshalled into the gate; the #1 Engine Flamed Out and the aircraft began to roll back wards. The Captain used normal brakes to stop the aircraft and set the parking brake. The flight crew called for a tug and continued the gate arrival under tow.

Narrative

We were holding short of X-XX Safety Zone waiting for Ramp Personnel to get in place. The Number 2 Engine was shut down; on the APU Bus; and the Parking Brake was set. Once Ground Personnel began to marshal the aircraft; I released the Parking Brake and moved the #1 Thrust Lever to get the aircraft rolling. Before the Engine spooled up to the thrust setting; there was an ENG FAIL Light on the #1 EGT Gauge and the Engine rolled back/flamed out. With no Thrust; the aircraft began to roll backwards. I stopped the aircraft with Manual Braking and set the Parking Brake. I verified with the First Officer that he was seeing the same thing I was seeing regarding the #1 Engine Indications. He agreed and I moved the Start Lever to Cutoff at that point. There were no other abnormal engine indications or any other indications in the flight deck related to the engine failure that we could see. I called Ops for the tug to pull us into the gate and called Maintenance to meet us. The aircraft was already scheduled to terminate in ZZZ. I made a logbook entry concerning the incident and once the Mechanic boarded the aircraft; I briefed him verbally on what had occurred.