A few clarifications, because the comments do not state clearly (except the last one) whether we are talking power to the
module or to the
coil. It matters and is part of the general confusion about the PerTronix systems. If in-doubt, see
post #3. Also note that "resistor", "ballast", and "pink wire" are functionally the same thing.
Have run both Pert I and III modules and always left the pink wire in circuit with no negative effects.
That you are aware of.

If you mean a resistor to the module, then the answer is never, as explained previously. If you mean resistance to the coil - while the P-I should be run with the resistor to the coil (see
post #3), the P-III should not. PerTronix even states that it will work (run), but that the spark power will be much reduced, and the multi-spark feature will not function. A lot of money to have it work like a P-I with a rev-limiter!
Pertronix always recommends bypassing the pink wire but they don't actually say it won't work.
You mean to the module? They don't say it won't work, because most of the time, it will seem to work fine. Seem to. I know (and have proven several times) otherwise, with full explanations as to why. PerTronix is caught between giving the impression it is a "drop-in" system, yet saying that it is to be wired without resistance to the module - complicating the process. Not really a drop-in when done properly. They do not mention how your engine will quit quickly if the alternator stops working on the way home, or clearly tell you how you may not get your engine started with a low battery - if the resistor is left in-line to the module. I believe their instructions are a bit misleading, although they do actually tell you to do it a certain way if you keep reading, and I understand they are also in it a profit business. I have no issue with profit. Just the unclear part. I will also repeat what you said - "Pertronix always recommends bypassing the pink wire...". In any case, I will say that wiring it as in
post #3 will make it run correctly, better, longer, and more reliably than other methods.
Stock ford coils and any other coil requiring a ballast, AFAIK, should still be run with the pink wire to avoid damage to the coil.
With a P-I, yes. With the other versions, no. See
post #3, and explanations as to why in other posts. FWIW
David
The pics were taken directly from PerTronix-I Ignitor directions. Center text and color added for clarity. Remember, no resistance to any module version, and resistance to the coil for P-I only: