I believe that some third party software have installed its kernel extension (likely for some hardware network device) to the /System/Library/Extensions/IONetworkingFamily.kext/Contents/PlugIns/ and have corrupted the signing of the IONetworkingFamily.
My current view of the problem is the next:
Long time ago (like in 10.6), the OSX required to install hardware network drivers to the IONetworkingFamily.kext/Contents/PlugIns/. This have changed since (as far as I remember) in OSX 10.8, but since all continued to work, that doesn't bothered drivers providers. Now OSX forbids this path for third-party network drivers.
There are two possible solutions: a) remove the software that corrupts the IONetworkingFamily and ask that company to fix the problem or b) disable network interfaces vnic0 and vnic1 on host (this doesn't affect most customers of Parallels Desktop).
As a workaround (a) delete the incompatible driver with the steps:
1) start the /Applications/Utilities/Terminal and execute the command
ls /System/Library/Extensions/IONetworkingFamily.kext/Contents/PlugIns/
On my machine it displays the next:
$ ls /System/Library/Extensions/IONetworkingFamily.kext/Contents/PluggIns/
AppleBCM5701Ethernet.kext AppleIntelI210Ethernet.kext AppleYukon2.kext Intel82574L.kext nvenet.kext
AppleIntel8254XEthernet.kext AppleVmxnet3Ethernet.kext IOEthernetAVBController.kext mDNSOffloadUserClient.kext
If you aren't sure what to delete, post here the output, so I could say if there are anything wrong
For workaround (b) do the next steps:
1) Quit the Parallels Desktop
2) start the /Applications/Utilities/Terminal
3) execute command "cp /Library/Preferences/Parallels/network.desktop.xml ." (without quotes)
4) execute command "open /Applications/TextEdit.app network.desktop.xml" - it will open the Parallels Desktop network configuration in the editor
5) In all sections <ParallelsAdapter> change the value of field <Enabled> to be zero and quit the TextEditor, saving the file
6) execute in terminal command "sudo cp network.desktop.xml /Library/Preferences/Parallels/network.desktop.xml" - it will ask for the password. Please type it.
7) try whether it fixes the problem with networking in VM