Hi, take a look into the system log (/Applications/Utilities/Console.app). There could be an error "ERROR: invalid signature for com.apple.iokit.IONetworkingFamily, will not load".
If it is the case, then it means that some third party software is incompatible with El Capitan. Most likely it have copied its network driver to the improper location and broke the signing of the IONetworkingFamily package.
As a workaround either 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/
it shoud display only
2$ 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
post here the output, so I could say if there are anything wrong
2) another workaround is to disable some functionality of Parallels Desktop with 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