Parkinson's men opened the scoring through a Tony McMahon penalty but three unanswered goals from the Lions before the break eventually left City with a 3-1 deficit to try and overturn in the second leg at The Den on Friday.
After a frustrating opening half in front of a crowd of nearly 20,000, City dominated virtually all of the second period against Millwall, and Parkinson says his squad can take strength from the improved showing.
He said: "We were certainly disappointed with the nature of the first two goals. They were two soft goals to concede from our point of view.
"We are a team not used to conceding, especially in those situations. The two goals rocked us a bit and made us slightly edgy on the ball after starting the game so well.
"Second half though, we did a lot better. The lads showed a lot of character and courage to get on the ball and get us playing again.
"We spoke at half-time about trying to come out on top of a 45-minute game of football. I am pleased with our response after the break.
"We created a lot of scoring opportunities but we weren't able to take one of them. We also had a strong shout for a penalty turned down.
"We must take great heart from the way we played after the break. The tie is still not over.
"We have to go down to The Den on Friday believing we can get back into this game.
"I certainly feel we still have a great chance to do so.
"We will re-group in training this week and get ready for what we be an exciting match at The Den.
"We have done well on our travels this season on the whole and we know what is expected of us.
"If we play like we did in the second half today (Sunday), we'll have a chance."
City trio Billy Clarke (fractured cheekbone), James Hanson (calf) and Reece Burke (groin) all missed Sunday's first leg due to injury.
Parkinson added: "Reece Burke looks the closest at this moment to being involved next Friday.
"Billy Clarke needs a bit more time, and Hans has unfortunately not really made progress this week with his injury.
"We will be investigating that further. He will be having another scan on his calf on Monday.
"We want to get to the bottom of it because we are still no nearer finding out exactly what is causing this problem."