Как то так?
<?php
$headers = array(
'X-Parse-Application-Id' => ${APPLICATION_ID},
'X-Parse-REST-API-Key', ${REST_API_KEY}
'Content-type: application/json'
);
$data = array(
'username' => "cooldude6",
'password' => "p_n7!-e8",
'phone': '415-392-0202',
'postfield1' => $_POST['postfield1'],
etc...
);
... connect to curl ...
curl_setopt($ch, CURLOPT_POST, 1);
curl_setopt($ch, CURLOPT_HTTPHEADER, $headers);
curl_setopt($ch, CURLOPT_POSTFIELDS, json_encode($data));