I think they should. The assets are vested in the custodian trustee. The Charity Commission operational guidance (OG39) defines the role of custodian trustee as:-
"In round terms, the custodian trustee holds the title to all the property of the trust but is not involved in the day to day management of the trust. The powers and duties of a custodian trustee are set out in section 4 of the Public Trustee Act 1906."
Whilst the parish council doesn't have the right to dispose of these assets (which remain under the control of the managing trustees), the same could be said of many other parish council assets, e.g. war memorials.