If so, why is the diode resistance in parallel with the current source? My intuition tells me it should be in series
A resistor in series with an ideal current source still produces exactly the same current through its load regardless of the voltage across the terminals. So its behavior is exactly the same as the ideal current source with no resistor.
Since the current changes according to the voltage, that gives me more reason to believe that the resistance should be in series with the current source.
On the contrary, the resistor must be placed in parallel with the current source in order to have the affect of producing less output current for high voltage loads and vice versa.
the APD is not biased, but if it was then the current would vary with the applied bias.
For the record, the effect of bias on the APD current is highly nonlinear, and can't be modelled by adding a simple resistor to the model. A simple(ish) model of an APD might look like this:
simulate this circuit – Schematic created using CircuitLab
The function \$M(v)\$ gives the avalanche gain of the device. It is (as mentioned before) a highly nonlinear function, something like an exponential of an exponential function of the voltage across the device terminals.