For Query APIs like [Overpass API][4] or [XAPI][5] the bounding box will be too large. Therefore
the best way seems to get a planet extract of the UK from [Geofabrik][1] and run [Osmosis][2] [osmfilter][2] on it to filter for [leisure=park][3] and maybe name=*. However this will probably return a lot more objects than you are searching for.
[1]: http://download.geofabrik.de/osm/
[2]: http://wiki.openstreetmap.org/wiki/Osmosis
http://wiki.openstreetmap.org/wiki/Osmfilter
[3]: http://wiki.openstreetmap.org/wiki/Tag:leisure%3Dpark
[4]: http://wiki.openstreetmap.org/wiki/Overpass_API
[5]: http://wiki.openstreetmap.org/wiki/XAPI