I have no need to be anonymous with my web traffic. AnonSurf appears to “torify” all traffic. I have a virtual machine setup running Kodachi OS which torrifies all traffic, but also provides “free” VPN and encrypted DNS services.
For your issue, can you plug an Ethernet (or Wi-Fi adapter) to try getting Internet using the second network? If you were able to do that, you could remove/purge AnonSurf, and then install it again.
Hopefully a Parrot developer like @dmknght or someone who uses AnonSurf can tell you what might be wrong with your configuration. I know you’ve had this issue for several days.