ANSWER: After fiddling with my code I realized that port 2082 and port 2083 are the same except that 2082 has no https:// so I changed the port to 2082 and it worked!

CODE:

function createDomain($domain) {
    // your cPanel username
    $cpanel_user = 'User';

    // your cPanel password
    $cpanel_pass = 'Pass';

    // your cPanel skin
    $cpanel_skin = 'paper_lantern';

    // your cPanel domain
    $cpanel_host = 'example.com';

    // subdomain name
    $subdomain = $domain;

    // directory - defaults to public_html/subdomain_name
    $dir = 'public_html/user_site';

    // create the subdomain

    $sock = fsockopen($cpanel_host,2082);
    if(!$sock) {
        print('Socket error');
        exit();
    }

    $pass = base64_encode("$cpanel_user:$cpanel_pass");
    $in = "GET /frontend/$cpanel_skin/subdomain/doadddomain.html?rootdomain=$cpanel_host&domain=$subdomain&dir=$dir\r\n";
    $in .= "HTTP/1.0\r\n";
    $in .= "Host:$cpanel_host\r\n";
    $in .= "Authorization: Basic $pass\r\n";
    $in .= "\r\n";

    fputs($sock, $in);
        while (!feof($sock)) {
        $result .= fgets ($sock,128);
    }
    fclose($sock);

    return $result;
}

API 1 is now dead. Also passing your cpanel password through a non-secure connection (ie port 2082 instead of 2083) is a very bad idea. Next thing you know someone will have hijacked your cpanel account!

However, combining the codes given here for authentication and here for adding a subdomain, gives us the following script which seems to work just fine:

<?php 

$cpanelsername = "example";
$cpanelpassword = "**********";
$subdomain = 'newsubdomain';
$domain = 'example.com';
$directory = "/public_html/$subdomain";  // A valid directory path, relative to the user's home directory. Or you can use "/$subdomain" depending on how you want to structure your directory tree for all the subdomains.

$query = "https://$domain:2083/json-api/cpanel?cpanel_jsonapi_func=addsubdomain&cpanel_jsonapi_module=SubDomain&cpanel_jsonapi_version=2&domain=$subdomain&rootdomain=$domain&dir=$directory";   

$curl = curl_init();                                // Create Curl Object
curl_setopt($curl, CURLOPT_SSL_VERIFYPEER,0);       // Allow self-signed certs
curl_setopt($curl, CURLOPT_SSL_VERIFYHOST,0);       // Allow certs that do not match the hostname
curl_setopt($curl, CURLOPT_HEADER,0);               // Do not include header in output
curl_setopt($curl, CURLOPT_RETURNTRANSFER,1);       // Return contents of transfer on curl_exec
$header[0] = "Authorization: Basic " . base64_encode($whmusername.":".$whmpassword) . "\n\r";
curl_setopt($curl, CURLOPT_HTTPHEADER, $header);    // set the username and password
curl_setopt($curl, CURLOPT_URL, $deletedir);            // execute the query
$result = curl_exec($curl);
if ($result == false) {
    error_log("curl_exec threw error \"" . curl_error($curl) . "\" for $query");   
                                                    // log error if curl exec fails
}
curl_close($curl);

print $result;

?>

The result should be something like this:

{"cpanelresult":{"func":"addsubdomain","event":{"result":1},"apiversion":2,"module":"SubDomain","data":[{"reason":"The subdomain “newsubdomain.example.com” has been added.","result":1}],"preevent":{"result":1},"postevent":{"result":1}}}

To delete the subdomain, run this query through the above script:

$deletesub =  "https://$domain:2083/json-api/cpanel?cpanel_jsonapi_func=delsubdomain&cpanel_jsonapi_module=SubDomain&cpanel_jsonapi_version=2&domain=".$subdomain.'.'.$domain."&dir=$directory";  //Note: To delete the subdomain of an addon domain, separate the subdomain with an underscore (_) instead of a dot (.). For example, use the following format: subdomain_addondomain.tld

And to delete the directory, run this:

 $deletedir =  "https://$domain:2083/json-api/cpanel?cpanel_jsonapi_module=Fileman&cpanel_jsonapi_func=fileop&op=unlink&sourcefiles=$directory";

Noel's answer from 2018 more than likely won't work anymore, but if you've come here looking for how to use the Cpanel API to add a subdomain, start at this link to see the arguments accepted for the Cpanel API Ver 2 for the SubDomain module addsubdomain.

Below is an example that worked perfectly well for me.

$whmusername = "cpanel_username";
$whmpassword = "cpanel_password";
$subdomain = 'newsubdomain';
$cpanel_ip = 'IP_ADDRESS'; //ip of cpanel or your_domain.com
$domain = "your_domain.com";

$query = "https://".$cpanel_ip."2083/json-api/cpanel?cpanel_jsonapi_module=SubDomain&cpanel_jsonapi_func=addsubdomain&cpanel_jsonapi_apiversion=2&dir=/public_html/".$subdomain.".".$domain."/&rootdomain=".$domain."&domain=".$subdomain."";

$curl = curl_init();
curl_setopt($curl, CURLOPT_SSL_VERIFYPEER,0);
curl_setopt($curl, CURLOPT_SSL_VERIFYHOST,0);
curl_setopt($curl, CURLOPT_HEADER,0);
curl_setopt($curl, CURLOPT_RETURNTRANSFER,1);
$header[0] = "Authorization: Basic " . base64_encode($whmusername.":".$whmpassword) . "\n\r";
curl_setopt($curl, CURLOPT_HTTPHEADER, $header);
curl_setopt($curl, CURLOPT_URL, $query);
$result = curl_exec($curl);
curl_close($curl);

Note that if you want to see what Cpanel is returning as a response for $result, then place print $result; after curl_close($curl);