It appears from what is reported in this thread and wording on the notice that the estate use in the car park that they are under the misapprehension that the SSSI designation on part of the estate gives them the right to control the access to the area.
This quote from Nature Scotland publication
https://www.nature.scot/sites/default/files/2017-06/Publication%202011%20-%20Sites%20of%20Special%20Scientific%20Interest.pdf gives the case as it stands with regards to access.(Bold emphasis added)) This document is intended for landowners or occupiers that have an SSSI on their land.
Public access on to SSSIsSSSI designation does not increase or reduce public rights to take
responsible access (as described in the Scottish Outdoor Access Code),
nor does it increase or reduce the expectation on owners and occupiers to
allow responsible access. It is an offence for any person, including visitors,
to intentionally or recklessly damage the natural features of an SSSI. Please
tell us if you have reason to believe any person is damaging the protected
natural features of your SSSI.
Many people, whether they realise it or not, visit areas designated as SSSIs
for a wide variety of activities without disturbing the natural features of the
sites. We are always happy to discuss enhanced access with owners and
occupiers where this would not harm the natural features of the site.
Where visitors’ activities or numbers do cause damage, we will be pleased
to discuss with you and others how this might be addressed.
Closure of an SSSI for conservation purposesLocal authorities may remove land, including that within an SSSI, from
access rights for up to five days or, with the confirmation of Scottish
Ministers, for up to two years.