I am posting here as it got linked from access=permissive · Issue #5987 · streetcomplete/StreetComplete · GitHub
As I understand, there is desire to distinguish publicly run playgrounds and privately-controlled ones. This can be done using Key:operator:type - OpenStreetMap Wiki - and may require tricky research but also useful for some purposes. For example privately controlled-ones are more likely to suddenly have restricted access (access=private)
But municipal playgrounds may also be dismantled or become more restricted. Maybe city opened kindergarten in building next to one and turned it into one usable only by children going there. Municipal playground may be also removed because it collided with some construction or there were no funds to renovate dangerous and damaged equipment.
Note that in general playgrounds/parkings etc have no statutory legal right of access unlike right of way present in UK and some other places ( Access provisions in the United Kingdom - OpenStreetMap Wiki Rights of way in England and Wales - Wikipedia ). In such places it is illegal to block access to path/bridleways etc with such legal status. As far as I know is not really similar legal situation for German playgrounds. The same in other countries.
Therefore access=yes and access=permissive on playgrounds and parkings is effectively the same thing.
If you want to be extra-accurate it is fine to use access=permissive instead of access=yes on all playgrounds, but I would not expect all other mappers do the same. And using them interchangeable in this situation is not a mistake.
Disclaimer: I am not an expert on UK ROWs, I do not speak German at all.