Defamation is not limited to individuals. Companies can sue for libel, for example, as could an independent school, if they can show whatever has been put out there has caused or is likely to cause serious financial loss, which sounds pretty unlikely in this context, but is possible.
Government and governmental institutions (local authorities, state schools, NHS trusts etc) can't sue in defamation, but they can be sued.
There is absolutely no power to stop parents in general discussing the school, their own or other peoples' children, the staff, or anything else on social media. Blanket discussion bans aren't permissible.
Of course, any aggrieved person could sue a poster after something has been put on social media-in defamation, malicious falsehood, under the Protection from Harassment Act or any other civil cause of action they thought fitting. Really terrible postings could result in criminal action under the Malicious Communications Act or something similar. Posting something that discloses another's personal information without their permission could well be a breach of the Data Protection Act.
It still would not be a matter for the HT, unless she were the person who was suing.