Jones put the Bantams in front in the ninth minute when his corner from the left squeezed in at the front post.
James Hanson should have extended the lead only to miss with a header from close-range, while Cody McDonald squandered Gillingham's best chance in the second-half as they slumped to the bottom of the table.
The Gills are still awaiting their first win of the season while Bradford, also promoted last season, sit sixth and Parkinson believes it is nothing less than they deserve.
"I always said the cup games [last season] in front of the TV cameras and big crowds were going to be a great experience for our young players and they're now used to dealing with those occasions," he said.
"We've come up a level and we're going to bigger grounds, but it's not fazing them and they're enjoying the challenge of each and every game.
"Gary Jones epitomises everything that we want from the football club. He never gives in. He drives the team on and he's an outstanding player. He's come up a level, but he looks comfortable.
"We started in a very positive fashion. We looked a good side, dangerous, and could have scored more, but we couldn't find that second goal which was frustrating.
"You know when you win 1-0 away from home your team has had to show a lot of qualities and I felt we really did that."
- Sports & Recreation