Both the NAT and Host-Guest adapters are and stay connected (green) with and without the VPN connection established on the Mac side. Again, I can ping the network using IP addresses (numbers), but not names (www.example.com). So, maybe there's something wrong with accessing the DNS servers? I'd think that, I can't plug in an IP address (number) as a URL either... That would narrow the issue down to the VM not detecting or using the proxy server settings in the Mac's network preferences, right? That's set to auto proxy configuration using a proxy configuration .pac file. How can we get that to work?
Please, do some tests (may be once more) from Guest OS with VPN and Parallels Shared Network enabled:
1) ping www.parallels.com
2) ping 64.131.89.6
3) Switch off all proxies in Guest OS browser and put http://64.131.89.6/ as an URL
4) telnet 64.131.89.6 80