Here's a few (crazy) ideas if your only goal is to make-from-old... (also joined from SO just to post this).
If you're talented enough to program the system as low-level as the EEPROM/flash that the BIOS itself is loaded from, and still have reasonable control over the hardware:
You could in theory make a communication path via an available RAM slot, but absolutely nothing will support this, so you'd be entirely on your own (writing your own below-BIOS level drivers for chipset, CPU, RAM and your own custom kernel and operating system, plus the software you need to run). You'd require very sound knowledge in electrical engineering, programming, reverse engineering (nobody tells you how to interface with motherboard hardware directly, that's what the BIOS is for!). Not feasible in the end.
You could also communicate over PCI. This might actually work way better than my first crazy idea, but is definitely slower (though since both are theoretical, does that matter?). Again, it would require pretty good knowledge of electrical engineering (and RF now that I think about it; PCs are fast after all), and at the end of the day, the throughput probably won't be great, and certainly nowhere near supercomputer speeds.
I realize this means it's not feasible, and that was the original question, but I am the level of crazy to desperately want to try both after thinking about it for a bit, and thought I'd share anyway.
EDIT: If I knew my grammar was going to be corrected I wouldn't have posted :)