A minimalist WordPress plugin to allow child themes to inherit their parent's screenshot.

theme-screenshot-inheritance.php

<?php
/**
 * Plugin Name: Theme Screenshot Inheritance
 * Description: Child themes without screenshots inherit their parent theme's screenshot.
 *
 * Drop this in the /plugins or /mu-plugins directory.
 */

/**
 * Add parent screenshot to child theme.
 *
 * @param $prepared_themes
 * @return mixed
 */
function theme_screenshot_inheritance( $prepared_themes ) {

    // build list of names and id's
    $list = array();
    foreach ( $prepared_themes as $key => $theme ) {
        $list[$key] = $theme['name'];
    }
    $list = array_flip( $list );

    // child theme inherits parent screenshot
    foreach ( $prepared_themes as $key => $theme ) {
        if ( $theme['parent'] && !$theme['screenshot'][0] ) {
            // find theme in our list
            $parent_theme_id = $list[$theme['parent']];
            // get screenshot
            $parent_screenshot = $prepared_themes[$parent_theme_id]['screenshot'];
            // replace child theme screenshot
            $prepared_themes[$key]['screenshot'] = $parent_screenshot;
        }
    }

    return $prepared_themes;
}
add_filter( 'wp_prepare_themes_for_js', 'theme_screenshot_inheritance' );