Season 2 Episode 16 "Bart Dog Gets an F" in which Santa's Little Helper is goes to obedience school after destroying the family quilt
In this scene Lisa stays home from school with the mumps and Marge teaches her how to sew. The gif is from the part where Marge tells Lisa she just needs to "build up a callus" to sew and demonstates by poking her own finger with a needle then holding it over a lighter
Currently, mmr is about 86% effective against mumps. It may have been less so in the 80s. Also, she may have only had one round of shots. Could be she was just unlucky, and her immune response wasn't strong enough. It's not unheard of.