I think that in order to do this, you have to keep the other chip in reset to make the IO high-Z (not driven).
Usually the CS (chip select) does this, but there is no chip select available here. See reset as !CS (not chip select).
Update: This concept will not work for atmel parts. Keeping reset low only enters the programming mode. You could keep reset high on the part that you do not want to program. But you have to make sure the program does not use the SPI pins.
However, I would not go this route. I'd compare the costs of an operator changing the cable, using two programmers or using multiplexers.