Why do we calculate partial derivative of Error w.r.t output of Neural Network during backpropagation?
The partial derivative is used precisely because it separates concerns about how the value is calculated (from all the other parameters and outputs in the network) from how the value affects the output. This is purely by definition so you can do the calculation.

The "normal" derivative of the error function can be expressed as the sum of any "complete" set of independent partial derivatives. There are a few such sets possible - in a simple feed-forward network one for each layer plus one for all the weights combined. 

So in essence you are calculating the "normal" derivative w.r.t. the network weights, but due to the nature of the problem this is done by calculating the partial derivatives in multiple steps.



Caveat: I am probably not using those maths terms 100% accurately. For instance, I am ignoring the role of the training data and treating it as a constant.