First step was to make an adjustment at each rear wheel. This car has rear disc brakes, but the parking brake is independent and works like a miniature drum brake -- when you press on the parking brake pedal, shoes move outward against the inside of the rotor hub. Like most drum brakes, adjustment is done with a star-shaped wheel inside each hub. You remove one lug bolt and turn the wheel until the star wheel aligns with the lug hole, then you turn it with a screwdriver. (It's edge-on to the hole, so you can turn it by prying on the points with the screwdriver tip.) This is about as much of a pain in the butt as it sounds; you can't really see what you're doing, and it's hard to figure out which way to turn things except by trial and error. After a considerable amount of fiddling and a few false starts, I tightened them until they locked the wheels, then backed off a bit until the wheels turned freely again. So much for that adjustment.
Testing showed the parking brake now kinda, sorta held, but still went to the floor. So I dove under the car and tightened the adjustment under there, which effectively shortens the brake cable. It took quite a few turns -- I suspect no one had ever adjusted it, and the cable stretches over time -- but eventually the brake started to take after only a few clicks, and was fully on before the pedal hit the floor.
As a side benefit, while I was under there I finally found a loose exhaust mount that had been causing an annoying rattle for...well, pretty much since I got the car.