To add query fragment just put it inside the option array:
$node->content['links']['extra_link'] = array(
'#weight' => 10,
'#theme' => 'link',
'#path' => 'node/add/er-engagement',
'#text' => t('Add attendee data'),
'#options' => array(
'html' => FALSE,
'query' => array('field_er_cal_entity_reference' => $node->nid)
),
);
This will return an URL like this: node/add/er-engagement?field_er_cal_entity_reference=13