Enoch Showunmi pounced on a defensive slip to put Tranmere in front midway through the first half, but Miguel Angel Llera marked his return to the side with a 54th-minute equaliser for the Addicks direct from a free-kick.
Charlton had several penalty appeals turned down by referee David Coote and Parry was also baffled by some of the official's decisions.
Parry said: "It's a great result for us. I know Charlton are on the back of four straight defeats but they're a class team.
"I thought we were the better side in the first half. Charlton had plenty of possession, but they couldn't hurt us and our keeper only touched it about three times."
Parry added: "They had a few late penalty shouts and I'm not trying to impersonate Arsene Wenger, but I honestly didn't see them properly. I don't want to criticise the referees because I'm not paid that well."