jquery open page in a tab and pass some POST values

I would suggest creating an invisible form in your HTML like this:

<form id="invisible_form" action="new_window.php" method="post" target="_blank">
  <input id="new_window_parameter_1" name="parameter" type="hidden" value="default">
</form>

..and then submitting it via jQuery:

$('#new_window_parameter_1').val('value');
$('#invisible_form').submit();