Jsoup actually opens a new HTTP connection for each request, so your job is not entirely possible, but you can get closer:
Connection connection = Jsoup.connect("http://example.com");
Document doc = connection.get();
String authCode = doc.select("input[name='code']").val();
connection.data("id", "myID")
.data("username", "myUsername")
.data("code", authCode);
Connection.Response response = connection.method(Method.POST).execute();
HTTP ( GET POST), .
, (, Apache - HTTPClient), jsoup , , Jsoup.parse().