How to set parent-child relationship between differents custom post types

Finally I’ve found a working solution. The cartoon-series can be registered as you did but episodes custom post types can not be hirarchical (I think WordPress expects parent content be the same type as child content if the relationship is set using post_parent in wp_posts database table).

When registering episodes, the rewrite rule must be set to the slug you want, that is cartoon-series/%series_name%. Then we can filter the episodes link to replace %series_name% with actual name of the parent cartoon-series post type and a rewrite rule to say to WordPress when a cartoon-series post type is requested and when is a episodes.

add_action('init', function(){
    $labels = array(
        "name" => "Cartoon Series",
        "singular_name" => "Cartoon Series",
        "menu_name" => "Cartoon Series",
        "all_items" => "All Cartoon Series",
        "add_new" => "Add New",
        "add_new_item" => "Add New Cartoon Series",
        "edit" => "Edit",
        "edit_item" => "Edit Cartoon Series",
        "new_item" => "New Cartoon Series",
        "view" => "View",
        "view_item" => "View Cartoon Series",
        "search_items" => "Search Cartoon Series",
        "not_found" => "No Cartoon Series Found",
        "not_found_in_trash" => "No Cartoon Series Found in Trash",
        "parent" => "Parent Cartoon Series",
    );

    $args = array(
        "labels" => $labels,
         "description" => "",
        "public" => true,
        "show_ui" => true,
        "has_archive" => true,
        "show_in_menu" => true,
        "exclude_from_search" => false,
        "capability_type" => "post",
        "map_meta_cap" => true,
        "hierarchical" => true,
        "rewrite" => array( "slug" => "cartoon-series", "with_front" => true ),
        "query_var" => true,
        "supports" => array( "title", "revisions", "thumbnail" )
    );

    register_post_type( "cartoon-series", $args );

    $labels = array(
        "name" => "Episodes",
        "singular_name" => "Episode",
    );

    $args = array(
        "labels" => $labels,
        "description" => "",
        "public" => true,
        "show_ui" => true,
        "has_archive" => true,
        "show_in_menu" => true,
        "exclude_from_search" => false,
        "capability_type" => "post",
        "map_meta_cap" => true,
        "hierarchical" => false,
        "rewrite" => array( "slug" => "cartoon-series/%series_name%", "with_front" => true ),
        "query_var" => true,
        "supports" => array( "title", "revisions", "thumbnail" )
    );

    register_post_type( "episodes", $args );

});

add_action('add_meta_boxes', function() {
    add_meta_box('episodes-parent', 'Cartoon Series', 'episodes_attributes_meta_box', 'episodes', 'side', 'default');
});

function episodes_attributes_meta_box($post) {
        $pages = wp_dropdown_pages(array('post_type' => 'cartoon-series', 'selected' => $post->post_parent, 'name' => 'parent_id', 'show_option_none' => __('(no parent)'), 'sort_column'=> 'menu_order, post_title', 'echo' => 0));
        if ( ! empty($pages) ) {
            echo $pages;
        } // end empty pages check
}

add_action( 'init', function() {

    add_rewrite_rule( '^cartoon-series/(.*)/([^/]+)/?$','index.php?episodes=$matches[2]','top' );

});

add_filter( 'post_type_link', function( $link, $post ) {
    if ( 'episodes' == get_post_type( $post ) ) {
        //Lets go to get the parent cartoon-series name
        if( $post->post_parent ) {
            $parent = get_post( $post->post_parent );
            if( !empty($parent->post_name) ) {
                return str_replace( '%series_name%', $parent->post_name, $link );
            }
        } else {
            //This seems to not work. It is intented to build pretty permalinks
            //when episodes has not parent, but it seems that it would need
            //additional rewrite rules
            //return str_replace( '/%series_name%', '', $link );
        }

    }
    return $link;
}, 10, 2 );

NOTE: Remember to flush rewrite rules after saving the above code and before to try it. Go to wp-admin/options-permalink.php and click save to regerenate the rewrite rules.

NOTE 2: It is probably that more rewrite rules have to be added, for example to work for paginate posts. Also it may need some more work in order to have a complete solution, for example, when deleting a cartoon-series delete also all child episodes? Add a filter in admin edit screen to filter episodes by post parent? Modify the episodes title in admin edit screen to show the parent series name?