How to apply neural network parallelly on each part of image
The question has multi-fold answers:

First, you can simply crop the image 4 times and then pass them to the network. To parallelize you can use OpenMP. Of course the size of the input layer should match the size of the cropped patch. And just make sure your network doesn't use global variables. 

Next, I don't think it's a good idea to divide the image into parts. Another approach would be to adjust the network connections such that different parts are not connected in the first layers, but accommodate the information in the deeper ones. In short, the desired way should be encoded into the network, rather than some ad-hoc external procedures. This way your network can be trained end-to-end.