Jose Baxter scored the only goal in the 18th minute with his 20-yard drive being deflected past Latics goalkeeper Alex Cisak by team-mate Enoch Showunmi.
Parry said: "It was a big game and we have got to be pleased with the outcome. It was a fair result, especially considering our first-half performance.
"All the players put in a good shift. Everything is built from keeping a clean sheet. You don't want to be having to score three goals or so to win a game.
"It was a performance of a very high quality and I hope we can repeat it in our next outing. My biggest problem will be who to leave on the bench.
"It was interesting who got man of the match. It went to Lucas Akins but we thought defenders Ian Goodison and Ash Taylor were fabulous because Oldham are a handful."