Even though you can flight instruct using the privileges of a third class medical cert(which is what he was down to with a second class that old), flight instructors need to remember that second class medical privileges are needed to do the other kinds of operations that come hand in hand with working at a flight school such as photo flights, golf ball drops, and candy drops, etc....
It's best to just renew the second class every year while flight instructing. If only just to have it for the off chance that you might find yourself suddenly with the chance of getting you're first proper commercial turbine job. It's not like it's crazy expensive.