Comments: Approve when admin Replies, from the Front end

Well… All you have to do is to check what hooks are available during comment posting.

The one I think might be helpful is wp_insert_comment. So you could do something like this:

function my_wp_insert_comment($id, $comment) {
    if ( is_admin() && $comment->comment_parent ) {
        if ( wp_get_comment_status( $comment->comment_parent ) == 'unapproved' )
            wp_set_comment_status( $comment->comment_parent, 'approve' );
    }
}
add_action('wp_insert_comment', 'my_wp_insert_comment', 10, 2);

Just place this code in function.php file of your theme. This function will be called every time a comment is inserted into database. No matter how (from admin area, front end or by some plugin, etc.)