Garthorne Road Nature Reserve is owned by Network Rail IIRC and at least one local school I know of uses it regularly. Public access is restricted by the licences. There are rare species that need protecting there - it is possibly a remnant of Oak of Honor Wood.

As for other parks on that side of the tracks, what about Blythe Hill fields?

Am not sure if the reservoir under the golf course would support the weight of Lund Towers but I won't stand in his way.