Why isn't the spark gap lighting the pilot on my furnace?
I finally found some time yesterday to take a closer look at the furnace.
Gas Pressure
I started by measuring the incoming gas pressure using a gas pressure gauge. I found it to be 6 WC (inch water column), which is well within the acceptable range specified on the furnace.

Dirty Gap
Next I took a closer look at the spark gap itself, and watched closely as the spark was generated.  I even recorded the spark with my video camera, which allowed me to zoom in fairly close.  This is when I noticed a bit of carbon buildup on the electrodes, and wondered if it could be that simple...

Cleaning the Gap
I grabbed a bit of 220 grit sandpaper, and cleaned the electrodes up a bit. steel wool is probably the tool of choice for this job, but I didn't happen to have any on hand at the moment (not yet unpacked).
Success!
After testing the pilot ignition a few times, the problem seems to be fixed. We had a cold night last night, and the furnace fired up fine all night. As I write this, the furnace has just cycled successfully yet again.

Update
I found this YouTube video from Grayfurnaceman that describes the problem exactly, and says that the only solution is to replace the assembly. I'll be replacing the pilot assembly once the weather warms up, so we'll see if that solves the problem.