This conundrum is happening at my own place.

The gas furnace has an electric igniter. When the thermostat triggers it, it makes the rapid sparking noise just like your basic stovetop and other furnace igniters. But, if the stereo is on, the clicking comes out of the speakers.
I checked grounding at both the stereo and furnace.
I moved the stereo to the opposite leg of my service.
I tried two fancy audiophile filters (thankfully money-back guarantee!)

I've run out of ideas apart from an expensive isolation transformer.
What have I missed?

- Eric