A strategy for disciplining children seems to me to be one of the things you would discuss before starting the job. Did you have this discussion before you started? You seem to know how to do it and you are a professional nanny so I can only assume the parents are not fulfilling their side of the agreed deal.

They have already undermined you in front of the child so I think you are now at so much of a disadvantage that without a radical change in the parents' attitude (which is highly unlikely, even if you do have it out with them) you are doomed to be frustrated and irritated with the child and your job.

The only solution seems to be to find a new family who share your philosophy and make sure you all know the rules going in.