I don't think you can necessarily conclude that parents are being alienated by the suggested action, nor that the perceived professionalism will be undermined. There will be many, many parents who fully support the teachers and their actions.
I am genuinely puzzled by attitude of the head teachers. I am aware of many of the problem that face teaching today but as far as I can see academy status would give heads more power to deal with them.
To use an analogy I don't think junior doctors have alienated patients nor has their professionalism been undermined by their recent actions - quite the reverse, they are seen as standing up for patients and the NHS against the government.
Apparently a very different situation (unless you take the view that industrial action generally is being whipped up by the far left)
Junior doctors have enjoyed public support so far, thought I suspect that 'good will' is wearing a bit thin for those who have had operations postponed.
According to the BBC (not generally considered a far right organisation) - the only area of disagreement between the two sides is payments for Saturday working. So when the doctors complain that the new contract would force them to work excessive hours either they are lying or they would be quite happy to work excessive hours if they had a bit more cash.
They are NOT striking to protect patient safety, they are striking for money.
Once the public realise this simple fact, they will start to become very isolated.