Post to inherit custom category background image from parent

You can use this function to output the parent category ID and then style off of that.

<?php
function wpse_74737_category_top_parent_id($catid) {
    while ($catid) {
        $cat = get_category($catid);
        $catid = $cat -> category_parent;
        $catParent = $cat -> cat_ID;
    }
    return $catParent;
}

And then call this in your body class: <?php echo wpse_74737_category_top_parent_id( $cat->term_id ); ?>