Because video playback software is based on lineair stream reading, this will be a hard to find option, because it requires parallel processing. The only applications I can think of that support these functions for consumer use are non-lineair video editing solutions.
I suggest you use a not too advanced non-lineair program like Kdenlive for Linux and Mac OS or Adobe Premiere Elements for Mac OS and Windows.
What you want to do then is open up 2 video tracks, both with the same video. This should be the video you want to watch of course. In most editing software this will automatically add a audio track for each video track, so you'll have 2 video tracks and 2 audio tracks. Disable one of the video tracks and select the desired stream for each audio track. In most recent versions a simple right click on the track and opening up a track's properties will do the trick most of the time.
Edit the project's settings to your desired output, because this might influence the project's preview output quality. Output quality is probably individually configurable though.
Before you hit playback, make sure that you open up a audio mixer panel / window, so you can play with the volume while playing back. Not all software editors will support this, but I know that Premiere does.
It's not a pretty solution (and you'll misuse the video editors) and probably some higher PC requirements might apply. Of course you might also consider to render the project and watch the output file on a video playback device on an external device or with MPC / KMP / VLC or whatever suits your needs, but this will prevend you from individually alter the volumes of the two (or more?) audio tracks merged into one. Because that's what happens: you'll mux or merge two audio streams into one.
While I was typing above answer I thought of another solution. Have you tried using macros to simultaniously click or hit the play button when you opened up two instances of your favorite program? You could try that out and disable or ignore the video of one of two instances.
Well I hope this helps you out! Good luck!