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<td width="68"><a href="directory_update.php?updateid=<?php echo $row_Directory['Sno']; ?>"><img src="images/btns/delete.png" width="36" height="36" border="0" /></a></td>

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mysql_select_db($database_Connection, $Connection);
$query_Directory_update = "update directory set Sno=<?php $Directory_update['Sno']; ?>where Sno='$_GET[updateid]'";
$Directory_update = mysql_query($query_Directory_update, $Connection) or die(mysql_error());
$row_Directory_update = mysql_fetch_assoc($Directory_update);
$totalRows_Directory_update = mysql_num_rows($Directory_update);

Parse error: parse error, expecting `T_STRING' or `T_VARIABLE' or `T_NUM_STRING' in C:\wamp\www\prophp\directory_update.php on line 84

$query_Directory_update = "update directory set Sno=<?php $Directory_update; ?>where Sno='$_GET[updateid]'";

try to modified the above code into this:

$query_Directory_update = mysql_query("update directory set directory.Sno='$Directory_update[Sno]' where Sno='".$_GET[updateid]."'");

I think you don't have any PHP error with that query, but I'm not so sure if the query successfully update the record, since I don't know what is the exact variable from your database and PHP code itself.