For what it is worth. Here is how I got the engines started from the 707-300 Cold-n-Dark (KSEA) Flight.

Flight Engineers Station
Battery (master switch)...................................................ON
DC Meters Selector.........................................................TR3
Essential Power Source Selector.......................................EXT PWR
AC Paralleling Selector....................................................External Power
Engine Bleed Air Switches................................................ON
Wing Valve Switches.......................................................OPEN
Left & Right Air-Conditioning Unit Pack Switches.................Both OFF

Overhead Panel
Ground Start Selector Switch............................................HIGH PRESS/LOW PRESS⁺
Engine Start Levers.........................................................START
Engine Start Control Switch..............................................GRD<sup>*

*</sup>Repeat for remaining engines.
⁺Sometimes it works on High Press and other times Low Press. Weird! :-/

I guess you can start the engines in whatever order you like but I read in QID 203 (from the Support KB) that the order is #3, #4, #1 then #2. Like Tim says, the engines only start if the Engine Start Levers are in the START position before turning the Engine Start Control Switch to GRD (this goes for ALL the engines). Thank you very much for pointing this one out Tim!

Hope this helps.

Mark