a playground for central Oxford?

Central Oxford lacks any kind of playground, or even any kind of public "run around" green space. Twice I have been stopped by tourists and asked where they can take their children to play. Lots of residents tell me they would spend more time in the city centre if they had somewhere they could let their kids run around in between shops or museums.

the three nearest playgrounds are circled; the ones identified by Open Street Map at the Story Museum and inside Magdalen College School are not public

The nearest public playgrounds are quite some way away. From Carfax, Friars Wharf is 900m, Great Clarendon St 1.3km, and Angel and Greyhound Meadow 1.6km. All of these require dealing with at least one main road; none of them have public toilets or baby-changing facilities. The grassy area outside the Natural History Museum offers some "run-around" space, but that's still a kilometre from Carfax.

In this regard Oxford (and most UK cities) compares badly with many European cities, where even much more constrained city centres, with higher population densities, find the space to squeeze in play areas and equipment, even if it is just a "micro playground" with one or two items. (One example: Göttingen's inner city has three decent play areas within about 300 metres of its central "girl and goose" statue.)

So any redesign of Oxford city centre should include some play areas.

• The most obvious location for a proper playground is St Giles. There is a lot of space available there, potentially even without removing that much of the car parking. A playground would fit in with the annual St Giles Fair where it would offer something for the families that can't afford the rides. Pounds Park in Sheffield offers an example of what can be done if there's the will to reallocate space.
• It should be possible to fit at least a small play area into Broad St, especially if it can be designed without needing to support turning buses.
• The space in Frideswide Square is underutilised and often feels quite "dead": it might be possible to fit in a few pieces of either play or sports equipment, or even a small fenced play area. This wouldn't be that useful for the city centre (the walk along Hythe Bridge St is pretty horrible), but would be great for people using the railway station.
• There are possible spaces in Bonn Square or Pennyfarthing Place, but they are quite small. It would also be good to locate a play area away from the Westgate, to help rebalance city centre footfall back to the north and east.

And there are some options that would depend on private landowners rather than the councils:

• The area in front of the Natural History Museum (University of Oxford) could perhaps fit in some "landscape" play equipment.
• As suggested in the comments, bits of the churchyards of St Michael's and St Mary Magdalen's could potentially be opened up to the public, but there isn't really room for a play area and that might not be considered appropriate.
• Further out, University Parks (University of Oxford) could fit in a play area. And most of the space in Christ Church Meadow (Christ Church college) is flood meadow allocated to cows, but something could perhaps be fitted in along the western side. Either of these would be a fair way out of the centre, however.