Hmmm. Here's what happened with Babydoll. I was able to get the blinking to stop, when I adjusted the switch exactly right on the parking brake lever. (It rarely does line up right, but in this case I was able to get it to work.) Later, I had to pull the driver's seat to fix the seat mechanism, and in the process I broke off the connector that goes to the switch. So it's definitely an open circuit now, but the light keeps a-blinking.
This would bother me more if I actually drove the car, but it and its owner are 80 miles away.