I managed to fix this thanks to several of the comments in this thread. It sounds like the secret is to use the x64 .NET runtime and NOT the arm64 one (though you can have both installed).
My machine: Macbook Pro with M2 Pro chip and 32GB memory Sonoma OS v14.0
VM: Parallels 19 with Windows 11, 8 cpu cores 16gb ram
ArcGIS Pro 3.1.3
It was not working for me until I installed .NET Windows Desktop Runtime 6.0.24 x64 version. The x64 runtime must be doing the CPU virtualization needed by ArcGIS Pro (sorry if that is obvious, I'm still a n00b with virtualization environments). Microsoft says that .NET 7 is even better for this but I haven't gotten around to trying it just yet.