Parry scored after 15 minutes as City fielded a team that included new signing Jay Bothroyd and triallist, Nigerian international Wilson Oruma.
It was the Nigerian who scored City's second to earn the draw after the Bluebirds conceded two goals from Tony Onokah and Danny Smith who put the hosts in front.
Oruma has won an Olympic gold medal with Nigeria as has earned over a dozen caps with his national side.
"We'll give Wilson time to show what he can do, but we know he is a quality player," said manager Dave Jones.
"He has been voted African footballer of the year twice - and he has spent the last four seasons at Marseille.
"He hasn't played at all during the build-up to this season and hasn't been through a full pre-season training programme.
"Wilson has had one training session with us and then we were given clearance to play him against Chasetown."
Peter Enckleman, Peter Whittingham, Riccy Scimeca as well as Paul Parry all stared the game along with Bothroyd and Oruma.
On new signing Bothroyd Jones said
"He's got a lot of ability, you can see that and I think he's going to be a fantastic addition to us.
"He's got movement and knowledge, has some good silky touches but he's now got to settle down and score goals."
The draw at Chasetown means City end the pre season unbeaten.