Parkinson's men lost their third round encounter to MK Dons at stadiummk thanks to efforts at the beginning and the end of the tie by the hosts' on-loan Arsenal striker Benik Afobe.
The 2-0 scoreline to the Dons perhaps doesn't tell the full story of the tie, however, as City's men more than held their own for the majority of the match and probably looked the more likely scorers going into the final quarter of the game.
Parkinson said: "I think from the fifth minute to the eighty-sixth minute, we were probably the better team. It was a good display from us tonight (Tuesday).
"Our approach play was good but we just couldn't get that goal. Goals are obviously important in football though, and Benik Afobe has got two that have separated the teams really.
"When you look at the goal record they (MK Dons) have recently and you concede early, you do get slightly concerned, but the way we approached the game after that was excellent.
"We had so many good periods in the game - our wide players looked a threat in particular - but we just couldn't find that one moment to get us back in the game.
"There are slightly more acceptable ways to lose a game of football and tonight I think we have lost in one of the better ways in terms of the spirit, commitment and endeavour we showed out there. I feel the lads gave everything.
"Given our recent history in this competition, the lads knew they had to wear the shirt with pride tonight, and I felt they did do that. Obviously we didn't want to go out, but I don't think we could have asked for much more from the players on the night."
With Stephen Darby (tight groin) and Jordan Pickford (ineligible) joining Andrew Davies, James Hanson, Alan Sheehan, Lewis Clarkson and Matthew Dolan on the sidelines for the game, Ben Williams was given the keeping duties for the night and Gary Liddle moved from midfield to right back.
Liddle was also made captain in Darby's absence. In attack, Oliver McBurnie played up front on his own with Billy Clarke just behind him and Filipe Morais and Mason Bennett, who was given permission by Derby County to play, on the wings.
Parkinson added: "Gary (Liddle) slotted in really well at right back.
"With Stephen (Darby) out, we did think of putting Chris (Routis) out to right back and putting Gary in central defence, a role he is perhaps a bit more comfortable in, but I decided to keep the partnership of Rory (McArdle) and Chris there.
"It proved the right decision because I felt Gary did a good job at right back.
"Ben (Williams) has got another game under his belt tonight, which is another bonus for us. He was very good and assured during the game.
"The two wingers (Filipe Morais and Mason Bennett) played with a lot of energy. That role is normally a physically demanding one but they both coped well with it.
"We took both of them off in the end but they were excellent overall."