I am given the information that an air parcel undergoes isobaric heating from 0° C to 20° C, and that's all I'm given. I have to determine the work done by the parcel on its surroundings. I know that $dW = pdV$, and that $pV = RT$ ($V$ is the specific volume), but I don't know how to go about solving this in terms of the change in temperature and not the change in volume.
That's easy. Since it's isobaric the pressure is constant and the work is just $P\Delta V$, where $\Delta V$ is the total volume change. You just need to calculate the change in volume from the change in temperature when $P$ is constant, and you can get this from the equation $PV = RT$ that you wrote in your question.