select * from persons where (personLastName='Doe' and personBirthYear=1980)
or (fatherLastName='Doe' and fatherBirthYear=1980)
or (motherLastName='Doe' and motherBirthYear=1980)
$personQuery = array('personLastName' => $lastName, 'personBirthYear' => $birthyear);
$fatherQuery = array('fatherLastName' => $lastName, 'fatherBirthYear' => $birthyear);
$motherQuery = array('motherLastName' => $lastName, 'motherBirthYear' => $birthyear);
$query->where($personQuery);
$query->orCondition($fatherQuery);
$query->orCondition($motherQuery);
// also tried this
$query->where(array(
$personQuery,
$fatherQuery,
$motherQuery
), xPDOQuery::SQL_OR);

This question has been answered by Jako. See the first response.
$query = $modx->newQuery('whateverClass')
$c->where(array(
array('personLastName' => $lastName, 'personBirthYear' => $birthyear),
array('OR:fatherLastName' => $lastName, 'fatherBirthYear' => $birthyear),
array('OR:motherLastName' => $lastName, 'motherBirthYear' => $birthyear)
));
$c->prepare(); die($c->toSql());