Park deer are classified as wild animals, albeit within an enclosed area, and therefore can be culled and processed in accordance with the same legislation that applies to wild deer. However, they must be killed by hunting (ie, stalked and shot with a rifle). Also, they should not be handled, medicated, eartagged or fed concentrates although a lot of parks do, which is wrong and should result in them losing their park status. The only management intervention should be the provision of supplementary feeding of hay, silage or roots during severe weather, for welfare reasons.
Park deer can only be culled during the appropriate season for the species.
Farmed deer are domestic livestock. They can be handled, housed, medicated, fed etc, and traceability rules apply re movements between holdings and so on. Generally farmed deer need to be killed in an abattoir, but in some circumstances they can be killed on farm and transported to an abattoir for further processing.
Farmed deer can be killed at any time of year.
There should be a very clear distinction between the two management systems, but in reality the boundary is blurred by many grey areas.
The chap who can tell you a lot more about this than I can is
@cyberstag.