Yes if you update the theme any changes you have made to the files themselves will be over-written.
Which is why it’s often useful to use a child theme in order to make changes which will not be affected with parent theme updates.
We include a child theme for Parallel in Appearance > Setup Integral > Child Themes tab.
Can you provide a link to your website so I can have a look?
I checked out the demo of Integral and I do not see an issue with the background image in the testimonials section.