Can you use sparkling water to brew coffee? Has anyone ever tried it yet? What flavor/mouthfeel/extraction benefits are there, if any?
If you heat the water as per most brewing methods, it would no longer be carbonated/sparkling by the time you brewed with it.

You would thus be confined to a cold-brew method, and if you wanted the water to remain carbonated you'd have to cold brew in a sealed container. That would be moderately inconvenient but should be possible. 

Or you could brew coffee by any means and force carbonate it after the fact, which would thus have no effect on the brewing.

Never tried it, don't find the idea personally interesting, but all it should cost you to try would be a bottle of sparking water and the hassle of getting the coffee grounds into it, waiting a day or so, and then pouring it out through a filter and cleaning up the mess before recycling the bottle. Please report your results.