The resistor pulled high between Reset and 5 V, nor a 10 µF capacitor between Reset and GND worked for me. But this did... I put a 10 kΩ resistor between Reset and 5 V, AND, I put a 0.1 µF capacitor between my Uno's pin-10 and the target chip's Reset pin.
Pin 10 is the slave reset pin used by the menu File → Examples → ArduinoISP sketch. For an exceptional site covering stuff like this, check out http://www.gammon.com.au (this is where I got the capacitor idea). My configuration is an Arduino Uno rigged to a ATmega328P-PU on a breadboard (with clock, etc.), with the lines/circuit noted in comments near the top of the ArduinoISP sketch (plus the capacitor and resistor).
Arg... I can't seem to delete my comment here and re-editing is not working. Anyway, got faked by the upload, still failing at burnBootloader... I will update if I get this going...
Entry later in the day: I got the ArduinoISP sketch, running on a Uno R3, to burn bootloaders onto a slave ATmega328P chip on a breadboard. It didn't need capacitors or resistors on the Uno's reset pin.
Ultimately, I found burning bootloaders with this method unreliable. It only worked once in a while. I ended up using OptiLoader. The OptiLoader sketch worked every single time! Never an issue. The source code is clean. The dude that wrote it knows what he's doing, no doubt.
BTW: I ignored pin 9 for power and supplied my own. So in the end I'll be using OptiLoader to burn boot loaders. I'll use the ArduinoISP sketch really only for uploading normal sketches. It seems to be fairly reliable for that, but not for burning bootloaders (at least in my experience over the last day).