Do this. It works the same in Parallels as on bare metal. First confirm your Debian VM is x86_64. Edge for Linux is built for amd64. If your VM prints arm64, it will not install.
Run:
dpkg --print-architecture
If the output is amd64, set up Microsoft's Edge repository and install:
sudo apt update
sudo apt install -y ca-certificates curl gnupg
sudo install -d -m 0755 /etc/apt/keyrings
curl -fsSL https://packages.microsoft.com/keys/microsoft.asc | gpg --dearmor | sudo tee /etc/apt/keyrings/microsoft.gpg > /dev/null
sudo chmod a+r /etc/apt/keyrings/microsoft.gpg
echo 'deb [arch=amd64 signed-by=/etc/apt/keyrings/microsoft.gpg] https://packages.microsoft.com/repos/edge stable main' | sudo tee /etc/apt/sources.list.d/microsoft-edge-stable.list > /dev/null
sudo apt update
sudo apt install microsoft-edge-stable
This adds Microsoft's signing key in the modern keyrings location, registers the official Edge repo, then installs Edge. Future updates will arrive with apt upgrade.
If your Debian VM is arm64 on an Apple silicon Mac, Edge for Linux is not currently provided for arm64. In that case tell me your Mac model and Debian version, and I will suggest the best supported workaround for your setup.