Diode contribution in photo-diode's model
The diode in the model provides the regular diode current contribution, i.e. the exponential current flow that you expect if you forward bias any normal diode. Let's call the current through that ideal diode I_Dp. Then the current through the load (including the parasitics Cp, Rpp, Rsp) would be the sum contribution of ip+id+I_Dp, where ip is photocurrent, id is dark current, and I_Dp is the exponential current Is(exp(qV_A/kT)-1). In other words, I_Dp != ip+id. 

I think the confusion might come from the somewhat recursive-looking nature of the model. That or the notation of id for dark current.