I got it working. I was adding the authentication to the header incorrectly:
$apikey = '<api_key>';
$auth = base64_encode( 'user:'.$apikey );
$data = array(
'apikey' => $apikey,
'email_address' => $email,
'status' => 'subscribed',
'merge_fields' => array(
'FNAME' => $name
)
);
$json_data = json_encode($data);
$ch = curl_init();
curl_setopt($ch, CURLOPT_URL, 'https://us2.api.mailchimp.com/3.0/lists/<list_id>/members/');
curl_setopt($ch, CURLOPT_HTTPHEADER, array('Content-Type: application/json',
'Authorization: Basic '.$auth));
curl_setopt($ch, CURLOPT_USERAGENT, 'PHP-MCAPI/2.0');
curl_setopt($ch, CURLOPT_RETURNTRANSFER, true);
curl_setopt($ch, CURLOPT_TIMEOUT, 10);
curl_setopt($ch, CURLOPT_POST, true);
curl_setopt($ch, CURLOPT_SSL_VERIFYPEER, false);
curl_setopt($ch, CURLOPT_POSTFIELDS, $json_data);
$result = curl_exec($ch);
var_dump($result);
die('Mailchimp executed');