$client = new Services_Twilio('MyAccountSID', 'My auth token');
$client->account->calls->create($from_number, $to_number, $url_or_AppSID, array(
    'TwilioParam1' => 'value1',
    'TwilioParam2' => 'value2',
    'MyCustomParameter1' => 'CustomValue1',
    'MyCustomParamete2' => 'CustomValue2'
));

Map<String, String> params = new HashMap<String, String>();
params.put("From", myTwilioPhoneNumber);
params.put("To", customerPhone);
params.put("Url", myHandlerUrl));
Call call = client.getAccount().getCallFactory().create(params);
// THE LINE BELOW IS THE KEY TO PARAMETER PASSING
db.store(call.getSid(), myCustomParametersJSON);

Sting callSid = request.getParameter("CallSid");
// Optionally sleep 20 ms to make sure that data written by
// db.store(call.getSid(), myCustomParametersJSON);
// can now be read.
CustomData customData = parseJSON(db.fetchKey(callSid));
TwiMLResponse twimlResponse = new TwiMLResponse();
Say sayMessage = new Say(makeCustomMessage(customData));
twimlResponse.append(sayMessage);
...