Parsing: Entity Classes to DTO
There is this concerning GWT and GAE objects. If that doesn't work for you, I would rather have a class of public static methods that does the conversions rather than sticking it into some kind of "wrapper class" that contains object states. For example,

public final class JDOEntityUtil {

   static public AppointmentDTO toDTO(AppointmentEntity entity) {
       AppointmentDTO dto = new AppointmentDTO();
       dto.setId(entity.getId());
       dto.setFirstName(entity.getFirstName());
       dto.setLastName(entity.getLastName());
       dto.setAppointmentDate(entity.getAppointmentDate());
       dto.setNextAppointmentDate(entity.getNextAppointment());
       dto.setAttendant(entity.getAttendant());
       dto.setDetails(entity.getDetails());
       dto.setDNI(entity.getDNI());
       dto.setTreatment(entity.getTreatment());
       return dto;
   }

   static public AppointmentEntity toEntity(AppointmentDTO dto) {
       AppointmentEntity entity = new AppointmentEntity();
       entity.setId(dto.getId());  /* or however you want to construct this */
       entity.setFirstName(dto.getFirstName());
       entity.setLastName(dto.getLastName());
       entity.setAppointmentDate(dto.getAppointmentDate());
       entity.setNextAppointmentDate(dto.getNextAppointment());
       entity.setAttendant(dto.getAttendant());
       entity.setDetails(dto.getDetails());
       entity.setDNI(dto.getDNI());
       entity.setTreatment(dto.getTreatment());
       return entity;
   }
}