This is my personnel understanding…. some things are clearly open to some interpretation…
From what I can see all of these parks and open land come under Southampton Council, as you say, slapping up a sign has no legal standing unless backed up by a real Bylaw or a NFZ authorised by the CAA, both are not easy to obtain and need to show a very good reason.
The CAA regs say that all safety and operational rules are part of the licence and additional rules cannot just be added by an authority unless they can show good reason as above. Following all of the licence and drone code regs is what you need to do flying a UAV, and perhaps a risk assessment depending what you are planning to do.
What they can do is ask you not to take off or land on their land, but again it should be via a Bylaw or such, an ad-hoc sign just put up by the Council has no legal standing, what they cant do is prevent you flying over such land, the CAA control the Airspace, no one else.
So unless the Bylaw is specific (or NFZ) open land or parks are fine, it does say that in the CAA documentation, pending the notes above.
But of course as you say, “I must point out that I have no intention to flout laws and I have empathy with common interest regarding disturbance and safety of park users and local residence.” I agree totally, so I would go where you want within reason and if challenged ask to see any specific bylaw or such, the CAA say…
“CAA CAP 722.
All safety precautions ascertaining to any flight are inclusive of the authorised licence held.
NFZ or restricted areas formally designated by an authority or regulatory body should be able to identify the specific laws, regulations or bye-laws that empower it to regulate the use of an UAV, or more usually, the land from which they are operated, much as the CAA has set out the regulations that it applies above. We therefore recommend that if a restriction is imposed by a body that may legitimately apply to your flight, you request that information from the relevant authority or regulatory body “
Common sense, follow the code and have CAP 722 to hand… you cant go wrong… ha-ha.