As the questioner suggests, it is the ionization of the air that makes the path of the spark change. Stray radioactivity or cosmic rays will ionise air (neutrinos won't), but I suspect the most important source of ionisation here is the sparking itself: the first spark to discharge will produce ions all along its path, and these ions will be mixed in with the surrounding molecules, turbulently and unpredictably, by the currents generated by the spark's heating of the air. Of the ions produced by a spark, about half will recombine within 0.7 seconds, and about 90% after 6 seconds. So if the second spark occurs within a few seconds of the first, the spatial distribution of the ions produced by the first spark will be the chief factor determining the path of the second spark.
It would be interesting to see what happened if the device were allowed to rest for a few minutes in between each spark, so that the ions generated by the first spark would all have recombined before the second spark appeared.