Gauss's Law on Parallel Conducting Plates

I have a question regarding the application of Gauss's law to find the E-fields produced by two parallel conducting plates. My textbook (Halliday and Resnick 9e) states that there is no E-field above and below two parallel conducting plates; an E-field only exists between the two plates.

I don't understand this. Wouldn't the charge on the two plates still have an effect on the outside of the plate?

Speaking in terms of Gauss's Law, aren't there two ways to draw the Gaussian pillbox (see options 1 and 2 on the image below). In Option 1, $$q_{enc} = 0$$, which doesn't take into account the effect of the positive charge on the upper plate. In Option 2, $$q_{enc} = \sigma * A$$, which does take the positive charge into account and gives a non-zero E-field above the parallel plates.

How do you choose between Option 1 and Option 2 when calculating the E-field? Pillbox #2 does enclose a charge of $$\sigma A$$, however this is not because of a non-zero E-field on the top surface of the pillbox. This E-field is still zero, as indicated by pillbox #1. In this case, there is a downward E-field of $$\sigma/\epsilon_0$$ between the conductors (and hence on the bottom surface of the pillbox), which explains why there is a non-zero charge enclosed by the pillbox.