Got it answered over on Stack Exchange.
I quote from user s_ha_dum:
It sounds like you just need to remove support for the custom background color.
remove_theme_support( 'custom-background' );
You will probably need to hook it to get it to run after the parent functions.php
function disable_bg_wpse_97248() {
remove_theme_support( 'custom-background' );
}
add_action('after_setup_theme','disable_bg_wpse_97248',100)
I just added that function to my child’s functions.php and it worked like a charm.