Let's be clear. Feathering a (piston) engine means stopping the engine. It is an emergency procedure, when an engine is damaged or has run out of fuel. The process normally involves turning off the fuel, switching the ignition off, and feathering the prop. The last because, even though the engine is now officially dead, the airflow acting on the propellor blades will exert tremendous pressure to turn them. This leads to increased drag (the last thing you need when you have just lost an engine), and will probably lead to more damage to the engine if the reason for the failure is mechanical, rather than say running out of fuel. So feathering the propellor blades means they turn perpendicular to the airflow and the drag of the stationary blade is minimised. And it was a regular ocurrence in the piston engined world.
You do not idle an engine with a feathered airscrew. As far as I know no operational piston engine had a clutch mechanism to allow the engine and propellor to be disconnected.
This may not apply to the turboprop word though. The Double Mamba in the Gannet allowed one of the paired engines to be shut down in flight and decoupled from the transmission, and I have a feeling the Bregruet Atlantique MR aircraft was intended to patrol on only one of its two (uncoupled) engines, the other I believe being shut down (not idled) once it reached its patrol area.
I don't know, but I hypothesise that jets may idle an engine in similar circumstances, if the fuel cost of idling is less than the fuel cost of the extra thrust to overcome the drag arising from a shut down engine.
Parallel to the airflow?
PT6 engines always seem to start up with the propeller blades in the feathered position - which always seems odd to me.
Some light aircraft piston engines do actually have clutches - for starting I think.