The door that Yusupov refers to is shown in the first picture at the side of the house. It is partially hidden in the picture by the trees but you seen see it between the windows. It is 100% certain that this is the door although it is now metal and once was wood. Yusupov's references to it are quite clear;
it was in the courtyard, the side gates were open (or not) and the main gates would have been in the centre of what now is a children's playground.
You have to imagine an iron fence running from the side of the Palace on the left of the first picture across the front of the playground.
There was one set of gates (adjacent to the end of the palace) and one in the centre of the courtyard.
Yusupov, Pursihkevich and Lazovert describe it in their accounts of the murder. You will find references to the cat being parked in the courtyard infront of the door. the references to ingress and removing the body are very specific in both Y and P's accounts.
It was only when Rasputin is alleged to have escaped that Y says he ran with his butler, through the main entrance on the Moika and along the front of the fencing in a bid to cut Rasputin off
Richard