Capital One Cup quarter-finalists City had the better opportunities but could not break through a strong visiting defence.
Parkinson said: "It was a difficult game. I think we played some really good football, especially in the first half, but we couldn't unlock the door. A goal would have opened things up but Chesterfield did a really good job. They played well and made it difficult for us.
"Some of our football was very good and our discipline was also good, although we got frustrated in the second half and lacked a bit of calmness. But I couldn't fault the lads.
"Credit to Chesterfield, they are a team full of confidence following their 6-1 home win over Hartlepool in the FA Cup on Saturday."
Parkinson added: "I was especially pleased with Will Atkinson, Gary Jones and Zavon Hines - we played some really good stuff."