It is standard practice and good practice to use the smallest image possible and have the browser tile it via CSS.

If you mean the blue gradients at the side, you do have a tiny problem I'd fix, that's the background blue gradient has been placed on the body tag, overriding the style for the background image in the CSS. The CSS has a background image pointing to ../assets/images/background.gif yet the body tag in the page has style="background-image: url('./assets/images/background.png');" placed on it.
I'd remove that and replace the style in the CSS with that image.

As for the tileing, As it's being set as a background-image, it's presumed to need to be repeated automatically. To stop it repeating change

background-image : url( "../assets/images/background.gif" )With
background-image : url( "../assets/images/background.gif" ) no-repeat
But then you'll have to make that background image very long, 1600px or taller to make sure the end of it isn't seen by users with very high resolution screens.

As for them complaining it's tiled. That is the right and best way of doing it.  :-/ Maybe there's a misunderstanding in what they mean...