Conclusion:
Thinking that bpat1434's solution was a bit.. well on the alternative side, I decided to run two separate instances of mysql_fetch_array() instead.
I'm now doing this:
$adminquery = mysql_query("SELECT param FROM parameter WHERE name='siteuser'");
if (!$adminquery) die('<h3 class="warning">fetching admin name failed.</3> '. mysql_error());
$row = mysql_fetch_array($adminquery);
extract($row);
$admin = $row['param'];
$passwordquery = mysql_query("SELECT param FROM parameter WHERE name='siteuserpasswd'");
if (!$passwordquery) die('<h3 class="warning">fetching admin password failed.</3> '. mysql_error());
$row = mysql_fetch_array($passwordquery);
extract($row);
$password = $row['param'];
It's somewhat surprising (to me anyways) that this result is not achievable in a more scholastically "correct" way, with just one round of mysql_fetch_array()
But ok, I'm gonna mark this sucker "solved".
Once again, thanks you guys 🙂