I have one notebook which has two WiFi cards that require non-free firmware and thus won't work with Parabola. However those cards also provide Bluetooth and that does work.

After installing bluez-utils (using a wire), I was able to set up a network connection over Bluetooth using the instructions from this page:
https://www.blog.willandnora.com/2017/08/06/bluez-5-46-and-personal-area-network-pan-connectivity-with-rpi3/

It comes down to:
pacman -S bluez-utils (this requires a wire now, can be disconnected after this completes)
systemctl start bluetooth
bluetoothctl -> pair & trust
dbus-send -> connect
dhclient

It would be nice if bluez-utils could be provided on all live images and have some documentation/scripts in place to make connecting easy. This would allow wireless connectivity on a lot more hardware even albeit slow without resorting to non-free firmware blobs.