Is there any chance it could have been Robin Hill Country Park on the Isle of Wight? I've just looked it up, albeit in Anthony Smith's "Animals on View" so from the 1970s, and the description says:-
"The principal feature of Robin Hill is the degree of familiarity it permits between the visitors and the visited. There is, for example, a ten-acre 'contact' walk-through enclosure where its Deer, Wallabies, domestic animals and free-flying birds are far more accessible to us than in normal cages."
I'm sure I read bad reports of Robin Hill in the 1980s, so it must still have been on the go then. Of course the real place for contact with Deer was at the Lowther Wildlife Country Park in the 1970s! My father had a very entertaining dual with a Japanese Sika stag that kept butting him at our picnic table (in future years whenever he looked displeased we called it his "Deer face"), and I remember witnessing a young Formosan Sika stag kick a small boy to the ground. No lawsuits then, it was all part of a good day out!