Hi! I Found the solution of this problem. You can use this to start Parallels without sudo and it should work.
1. sudo nano /Library/Preferences/Parallels/network.desktop.xml
2. find <UseKextless>1</UseKextless> (value can be -1), and change the value to the <UseKextless>0</UseKextless>
3. if you can't to find this tag, then just create it under root tag <ParallelsNetworkConfig ...>here</ParallelsNetworkConfig>
Result: you can start/restart Parallels without sudo - network will work.