Seems BattleEye isn't working on Windows 11 ARM. It fails during load. I know this is probably more on BattleEyes side than Parallels, but I want to make sure you can know this is affecting many people. Maybe Parallels can more officially reach out to BattleEye to try to get a resolution on this stuff. Like what drivers are not working.
I know that Battle Eye detects if you are using a virtual machine and will refuse to run. Games such as Rainbow Six Siege have this occurrence. I really don't think you have many solutions other than doing some research on spoofing this type of software. Sorry I couldn't be more help.