Hi Libby - welcome to the boards. Good Question!
Many parents seem to turn to the PTO for these types of issues. Have they specifically asked for your help? It sounds like they are concerned about school board policy and, therefore, school board issues. My advice is to remain neutral.
Perhaps this group of parents has valid concerns. And the BOE probably has reasons for their position (funding, school building size, etc.) since you say:
The admin are doing what they can and are feeling battered.
Your own bylaws may answer your question for you. My group's bylaws, for example, clearly state that it IS NOT the PTO's role to interfere in policy:
The PTO shall not seek to direct the administrative activities of the schools or to control its policies.
Most PTO members are not professional mediators. Your intervention in this matter, if not to mediate, is then to take one side or the other. This has been the downfall of many well intentioned groups.
Let these parents organize into a group, define their concerns and take them through the proper channels (Admin, BOE, State BOE, etc.) If both sides remain calm and reasonable and work together to make the school better, then there is no reason for it to turn into a fight anyway.
Continue to do your work as a parent group so the children can benefit from your efforts.