Question for people familiar with the Parallella project [1]: I see the board design has an FPGA. If I buy a Parallella, can I take the Verilog source code in this article, run it through some toolchain, and actually execute it on the board?
You should be able to. The SoC used for the Parallella is a Xilinx Zynq-7020 which is available on various FPGA dev boards too.
The Parallella will use some of the FPGA for various glue logic, but the code in question either is or will be open sourced - they're being very good about opening up whatever they can (since their goal after all is to get people to build stuff that'll drive orders for their chips going forwards)
[1] http://www.parallella.org/board/