It was the eighth home match without a win for Bradford, who lost leading goalscorer Nahki Wells to neighbours Huddersfield in a B£1.2million deal 24 hours earlier.
But Parkinson, looking for "a proven striker" to replace the Bermuda international, said: "I was very pleased with the performance considering everything that has gone on this week and the injuries we have got - it was tremendous.
"We got off to a good start with an early goal, but I was disappointed with the goal we conceded - it was an avoidable one.
"We asked the players to put into practice what we had worked on during the week and they did that.
"We also asked them to play with spirit and togetherness and we did that. I am pleased with the way we performed.
"They play three at the back, which leaves space in the wide areas and we worked on getting the wide players on the ball.
"We knew we could hurt them down the sides of the pitch. We did that and on chances created we deserved to win the match."
Parkinson had a special word of praise for striker James Hanson, who put Bradford ahead in the first minute.
"James Hanson played really well," said Parkinson. "His hold-up play was great and in the last 18 months he has contributed to the success of Nahki Wells. He is very unselfish and that is a great quality to have in any striker."
Parkinson also praised Scottish Under-18 striker Ollie McBurnie, who partnered Hanson up front.
"Oliver worked hard, but it is tough for him at the moment," he said. "He is young and inexperienced, but he has got so much quality and he is going to be a really good player."
Bradford made a great start with their early goal. Hanson got his head to a left-wing corner from skipper Gary Jones and, as the visitors' defence failed to clear, the ball came back to the tall striker, who stabbed it into the net just inside the near post.
However, Bristol City equalised after 11 minutes when a long ball out of defence from Derrick Williams was headed on by Jay Emmanuel-Thomas into the path of Scott Wagstaff, who scored with an angled shot into the far corner from the edge of the penalty area.
The visitors took charge after their equaliser, but Bradford had a great chance to regain the lead a minute before the interval only for Garry Thompson to fire across the face of goal with just goalkeeper Elliott Parish to beat.
Both sides hit the post with long range efforts in an exciting second half, Emmanuel-Thomas for Bristol City and Jones for Bradford, while Parish dived full length to keep out a goal-bound header from defender Rory McArdle.
- Sports & Recreation
- Nahki Wells
- James Hanson