You can do
$query = "INSERT INTO student_project
(project_ID, lecturer_role, LK_ID, SK_ID, project_name, project_desc) " .
"VALUES (NULL, '" . $_POST['role'] . "', '" . $_SESSION['user_logged'] . "', " .
" '" . $_POST['students'] . "', '" . $_POST['project_name'] . "', " .
" '" . $_POST['project_desc'] . "');
INSERT INTO project_assess (project_assess_ID, project_ID, LK_ID)
VALUES( '$assessid', '$projectid', '". $_SESSION['user_logged'] ."');";
$result = mysql_query($query)
or die(mysql_error());
or you could rename your queries uniquely and do
$query_SP = "INSERT INTO student_project (project_ID, lecturer_role, LK_ID, SK_ID, project_name, project_desc) " .
"VALUES (NULL, '" . $_POST['role'] . "', '" . $_SESSION['user_logged'] . "', " .
" '" . $_POST['students'] . "', '" . $_POST['project_name'] . "', " .
" '" . $_POST['project_desc'] . "'); ";
$query_PA = "INSERT INTO project_assess (project_assess_ID, project_ID, LK_ID)
VALUES( '$assessid', '$projectid', '". $_SESSION['user_logged'] ."'); ";
$query = $query_SP.$query_PA;
$result = mysql_query($query)
or die(mysql_error());
Always making sure that there is a delimiting semi-colon between each insert.