Ingenuous! I wonder if a LTE base station (eNodeB) could be build similarly? That would really catalyze innovation in cellular network design.

Unfortunately, no. Currently there are no complete and ready to use open source eNodeB implementations. SDR hardware - UmTRX - should handle LTE. Fabrice Bellard's (closed source) LTE implementation was (preliminary) tested with UmTRX.

As for Parallella - I'm not sure. I don't know enough about it to tell if its ARM core can handle CPU load required by LTE.

There is Michael Ossmann's HackRF project: http://greatscottgadgets.com/hackrf/

But unfortunately, there doesn't seem to be any news about when they plan to ship.