What a strange way to achieve something relatively simple! Why did they set up the charity and has it now been dissolved? Did the charity raise any money from funding sources that could not have funded the council directly? The group of parents could have worked with the council to submit funding applications in the name of the council. They could also have undertaken local events and other fundraising on behalf of the council and paid the proceeds into the council's account.
But in answer to your questions, if the council gave the charity money to purchase equipment, the charity must pay the VAT. The charity must also pay the VAT on items purchased using funds raised from other sources. The council may not allow third parties to utilise its VAT exemption under any circumstances. The council could have agreed to purchase £10,000 worth of equipment to assist the charity in its work, as long as the ownership of the asset remained at all times with the council and no money passed between the two bodies. In this case, the council could have made a legitimate application to reclaim the VAT.
Furthermore, the trustees of the charity may have contravened charity law in selling thousands of pounds worth of shiny new charity assets to a related party for a quid.