Pilot reported ground handlers informed the flight crew that an orange safety cone had been ingested by the left engine during parking procedures for gate arrival.

Narrative

The landing and taxi were routine. We had wing walkers and a taxi guide for the parking; the space was clear and we taxied in. APU was inoperative so I secured the right engine while we waited for external power to be applied. External power was applied and we powered the left engine down. Ground personnel came up after shutdown to say the top part of an orange cone had been ingested while we were idling waiting on external power. The cone was previously damaged and the top was partially torn from the base prior to our taxi in. The ground personnel had the base of the cone in hand when he walked up the stairs of the jet way. It was one of the cones that gets placed in front of the engines after shutdown. The base of the cone had none of the orange top still attached. I saw parts of the orange top of the cone in the engine when I looked at it. I wrote the engine up for a FOD inspection. If the cone hadn't been damaged; it would not have torn free from its base and been ingested when we had idle power and it was properly placed while we taxied in. It was just a bad cone.