Note: I have no experience of attempting what is described below (neither am I a lawyer nor an accountant). The process may range from a "small bureaucratic hurdle" to a "complex legal nightmare". If it seems a plausible approach, you would probably be well-advised to reach out to others that have established CASCs for help and guidance.
According to this HMRC page the two ways a body can claim Gift Aid is if either it is a recognised charity or if it is a Community Amateur Sports Club (CASC). So one option may be to try and establish a CASC. I suspect that this is unlikely to be an easy process, but may be a more likely approach than trying to get the council to establish a charity.
The Register as a community amateur sports club (CASC) page on the HMRC site (very) briefly describes the steps; as you can see from their eligibility criteria, to register as a CASC, you would first have to create a "Sports Club" of some form that:
has a formal constitution
is open to the whole community and has affordable membership fees
is on an amateur basis
provides facilities in the UK
is managed by "fit and proper persons"
You would probably need the co-operation of the local council to allow the proposed sports club the use of the local park. One of the (several) requirements of becoming a CASC is that it must:
- not make a profit, unless this is reinvested in the club and spent only on promoting participation and providing facilities for eligible sports
So it could, in theory, be possible to spend money raised (through both membership fees and Gift-Aid-qualifying donations) on the improving the facilities of the park (tennis courts, bowling green etc.).
However, note that How to Register page mentions (among many other requirements) the need to provide "accounts from the last 12 months" and "bank statements from the last 3 months". It doesn't (as far as I can see) explicitly state that the club must have been in existence for 12 months before applying for CASC status (it might be possible to send only what you have), but be aware that you may need to establish the club – and let it operate under its own steam – for a period before applying.