Resource id #x indicates that you've successfully acquired a resource in PHP. As pointed out, that means that the SQL did return a result. Use a command line client or something like phpMyAdmin to execute queries directly so see the results.
In PHP world, you now have a resource in which you can operate on. The resource by itself is useless; you cannot print out a resource and expect anything meaningful off of it. You need to issue a fetch command to it to actually retrieve the data. Since you are using a custom class, we cannot provide you information on the fetching itself. Using just mysqli library directly, you'd have:
$sQry = "SELECT parent, MAX(`timestamp`) AS maxTimestamp FROM " . PREFIX . "replies WHERE type = 'ticket' GROUP BY parent";
if ($qry = $mysqliObj->query($sQry))
while ($aRecord = $qry->fetch_assoc())
printf("Max timestamp for parent %s is %s" . PHP_EOL, $aRecord['parent'], $aRecord['maxTimestamp']);
Actually, with this list above you may be able to call the method fetch_array on the db object. That's a bad way of injecting functionality into it though.