Traditional war memorials were generally funded by public subscription, which, in most cases, meant that there was no lasting body or fund for their maintenance. The War Memorials (Local Authorities' Powers) Act 1923 enabled local councils to maintain, repair and protect them, under specific defined conditions (only if vested in the parish, financial limit, county council approval etc). These conditions were relaxed in the Local Government Act 1948, which removed the need for the memorial to be formally vested in the parish council and allowed the updating with new names from subsequent conflicts (and correction of previous errors). The Parish Councils Act 1957 removed the requirement to seek prior approval from the county council for expenditure on war memorials.
None of the above legislation specifically gives a parish council the power to create a new war memorial, however the General Power of Competence allows the council to do anything not expressly forbidden, which could include a new war memorial if it satisfies the public benefit requirement. As far as I can ascertain, there is no legal requirement for war memorials to be funded from public subscription. That may simply be a misinterpretation of the fact that the government did not provide any funding for local war memorials and encouraged communities to do their own fundraising. A new memorial (or significant repair to an existing memorial) could be funded jointly by public subscription and parish council contribution, as well as from external funding through various grant-making organisations.
All of the above relates more to a traditional war memorial than to a memorial hall, garden, recreation ground or similar. In the case of your memorial hall, I suggest you locate the various legal documents that would have been drawn up at the time. Who owned the land on which it stands? Was it sold or leased to enable the building to be constructed on it? Who commissioned the building work? Was there a properly constituted memorial hall committee and if so, what happened to it? Is the community centre a charity? Is it registered at the Land Registry? You may find that the building actually belongs to the parish council. Try the county archivist and local newspaper archives in your local library if you know specific dates (e.g. foundation stone laying, topping out, opening ceremony etc.)
A million quid sounds like a lot of money to spend on a 74-year-old building. I trust that it has been spent wisely.