Nahki Wells and Ricky Ravenhill had twice put the Bantams ahead only for Tendayi Darikwa and Sam Togwell to hit back for the home side, in front of nearly 3,000 travelling City fans.
It leaves Chesterfield, ninth in the League Two table, four points adrift of seventh-placed Bradford with three matches remaining
"I thought Chesterfield started slightly better than we did but we got through that period and did really well towards the end of the first half and got the goal," Parkinson said.
"Second half, I thought we completely dominated but they scored from a corner at the end when Togwell got round the back and stuck it in. It's really frustrating to concede like that so it's a feeling of disappointment and frustration because we dominated and should have won.
"I was frustrated with the ref, there were penalty claims in the second half and how we didn't get one of them I will never know, especially a handball which looked to me a clear-cut penalty.
"There's an air of disappointment about us because we've just dropped two points but when we wake up in the morning we will be positive again because if we keep playing they way we are we've got a great chance."
- Sports & Recreation
- Sam Togwell