The Nerazzurri struggled to get into gear early on and were made to pay when Marco Marchionni slipped the ball through to Nicola Sansone who, with only Handanovic to beat, neatly tucked the ball into the bottom corner.
Inter started to find their rhythm midway through the first half and came close to an equaliser when Mateo Kovacic found Rodrigo Palacio with a pinpoint pass, and the Argentine rounded the keeper but couldn't get a shot away from a tight angle.
Moments later, Kovacic almost found a way through, but lashed his shot over the bar as the hosts continued to peg back the Parma defence.
They equalised 90 seconds before the break when Palacio scored from close range from Fredy Guarin's cross.
However, the home side were behind again in first-half stoppage time when Handanovic failed to control the ball, which rolled into the path of Marco Parolo who netted for his sixth goal of the season.
A win for Inter will help close the five-point gap to third-placed Napoli, while a victory for Parma would put them into seventh place ahead of Torino.