To determine exactly what will happen to an ascending air parcel, you need to know when the parcel will become saturated. You can determine this with a sounding diagram if you know the dew point temperature of the parcel along with its temperature and pressure.
Just as a parcel's temperature and pressure is represented as a dot on a sounding diagram, you can plot a parcel's dew point temperature and pressure as a dot on a sounding diagram. If you do this, you have all the information you need..
Suppose we consider our favorite air parcel again. Its temperature is 20 C, and its pressure is 1000 mb. Let's say its dew point temperature is 0 C. These points are marked on the diagram in pink. along with the saturation mixing ratio line for our parcel. Below and to the right of this line, the air parcel is unsaturated and ascends dry adiabatically; above and to the left of this line, the air parcel is saturated and ascends moist adiabatically. So our parcel starts at 20 C, becomes saturated at 760 mb and -3 C, and ascends moist adiabatically above that level, reaching (for example) -25 C at 500 mb.
See how this works? Let's try another air parcel. Suppose we take an air parcel at 850 mb with a temperature of 10 C and a dew point of 4 C. What temperature will this air parcel have if it ascends to 500 mb?
This is a good spot for an in-class demonstration. If you're reading this on your own, do your best with the text-based description below.
Start by finding those two points on the sounding diagram. If you have a paper diagram to work with, you would now draw two lines: a line parallel to the dry adiabats that passes through the temperature point, and a line parallel to the saturation mixing ratio lines that passes through the dew point point. The first line shows how rapidly the parcel will cool at first as it ascends. It continues cooling at that rate until it hits the second line: the line of saturation mixing ratio for the parcel. In this example, that occurs at about 770 mb and 2 C. Above that point, it would cool at the moist adiabatic lapse rate, so you would draw a line parallel to the moist adiabats to determine its temperature above that level. In this example, the parcel would have cooled to -17 C by the time it reached 500 mb.