I am not an expert on Parallels but what you describe sounds like a feature that tries to open documents on Windows with apps on MacOS.
Can you open Excel for Windows at all?
If so, you can then open a copy of the NodeXL template file. It is located in the NodeXL directory, which on my system is located here:
C:\Program Files (x86)\Social Media Research Foundation\NodeXL Excel Template
The file you want is:
"C:\Program Files (x86)\Social Media Research Foundation\NodeXL Excel Template\NodeXLGraph.xlsx"
Does that open in the Windows side when you drag and drop it into the Windows Excel window?