Well there's two methods you could use here:
1. more scripting!.
1b. more scripting!
1c. Ok this method would involve creating a small event listener function which listens out for text started & text stopped, which could be used to show/hide an interface (or scene object) containing the background image, as needed. My only issue with this method is that it will keep toggling the background on & off when changing between different display text actions parts.
2. Same as above but without any scripting. Create a scene object (each scene) or a single interface that will contain the background image for the displayed text. The next part will require a little work but it gives you absolute control of when it should or should not be displayed. As simple as using an action part to show/hide the interface or scene object, as needed. This way you could have it open without closing until you are ready to close/hide it.
In the upcoming release, interfaces will be allowed to stay visible during cut-scenes so I recommend adding the image into an interface, as it saves messing about creating a scene object for each scene.
Are you sure you don't want a slice of cheese instead?