In spite of my starve the troll pledge, the list sure came to life for a couple days. Guess not much I can do about it.
Yesterday my parents' heat went out. I went there to check it out. Lit a couple burners on the stove, for temporary heat.
Turns out the Peerless boiler was unheated. I checked two limit switches (thermal disks) which were OK. Wiggled a few wires, turned the power back on. The machine hummed and clicked, and the three burners came back on. After a while, heat started to flow to the house. Still not totally sure what was wrong. Will follow up if the heat goes out again.