To say something sparingly means to say something in an infrequent manner. When a tire blows out, people typically have what's called a spare tire in the trunk of their car to replace the blown out tire. We hear the word "spare" in "sparingly". This joke is playing on the word "sparingly" because we can hear "spare" in it, relating to a spare tire. 
One of the tires just blew out , Tom said sparingly .