The recording is pretty good. Some minor pops on the 'b's and 'p's, but if it is a demo, that's probably not worth worrying about too much.
There are some difficulties that I have, and here are some things you might like to consider:
Your girlfriend has a pleasant voice and good pitching, but it loses power on the lower notes, and it becomes virtually just air. Maybe playing it up a couple of frets instead of in C might give her a chance to put some strength into the bottom notes.
I would like you to try some different chords. There's nothing wrong with doing your own version, but the chords you are using, to my ear, are robbing the melody of its life and movement. For example, in the second line where she sings "what e'er I done", the melody note on 'done' is D, where you are playing a C. Playing G works better here. If you do want to be adventurous, you could try a D or even a Bb instead of the G. There are many places throughout the track where there seems to be a mismatch between chords and melody.
Here is an example, but there are many variations from which to choose.
The Parting Glass Chords by The High Kings @ Ultimate-Guitar.Com