Unfortunately, the losers are the children, not the parents.
Not just those children, either. All the other unimmunised children and adults in the general population who may be at greater risk of infection as a result.
Having thought about this, I can see several issues. One is the question of whether parents are wise to decide not to immunise their children, and another is the standard of professional behaviour that should be expected of medical staff.
There is a lot we don't know. What missed immunisations might or might not have been relevant, the nature of the suspected "fever", etc. We don't know if the nurse put the note in a place that parents and patients were not supposed to look, and whether they pried. Since in many jurisdictions patients and/or parents have the right to see medical notes, this may not matter much. What does matter is an issue of professional detachment. When a patient has views or a lifestyle that conflict with an attending professional person's ethical or other opinions, that person should endeavour to avoid being influenced. Abortion, blood transfusion, immunisation, whatever. Writing "loser" or "flake" or "baby murderer" or "libtard" or any kind of comment of that type on a medical record is very unprofessional, as the father rightly said.
We don't know if the parents gave the nurse an earful about immunisation because they have some extreme views, or whether they were just misled, anxious about possible danger and wanted to keep their kid as safe as possible. If the latter then the nurse showed, in my opinion, a certain lack of empathy of the kind which is essential in a healthcare professional (that word again!)