a) The road turning
b) The building
As you may of realised Sydenham Avenue no longer has these chained posts running each side of the road, but Hall Drive does.
Burnage Court is an 1888 rebuilding of Dunedin House, which was the white building seen in the distance on the Pissarro painting.
Sydenham Avenue, Dunedin House and Fair Lawn replaced Abbey's Farm in the 1850s/1860s. Sydenham Hall (on the right) remained until later. What intrigues me the most about all this is that a footpath went from the back of Abbey's Farm through Sydenham, Beckenham and Penge parishes, right the way to Old Cople Lane!
You can follow this route roughly down Sydenham Avenue, Lawrie Park Avenue, Crystal Palace Park entrance, former Cricket Ground, almost to the point of the Grand Central Walkway. Here you will come across a red boundary post sticking out of the grass that predates the palace! There looked to be some kind of lodge at this spot, so it was probably an important point in the lane to have a boundary marker. On the other hand, I may have my bearings wrong, and the footpath never intercepted Old Cople lane at the point of the boundary post.