City again fluffed their lines by relinquishing an interval lead gained by an early strike by James Hanson.
Goals from Luke Norris and an unfortunate Hanson own goal handed the Kent visitors the spoils - to the dismay of home fans who made their feelings known at the final whistle.
Some of the booing was aimed in the direction of Parkinson, but as he prepares his side for the visit to Oakwell to face Barnsley on Saturday, he said: “I’ve had plenty of plaudits from this club over the years but I’ll fully accept getting stick because we have got to be better.
“The fans have been absolutely terrific for me ever since I’ve been at the club but we tested their patience. I’ve got to be honest about that.
“You can’t expect them to stay behind a team that has had a soft centre like we did. That’s not good enough.
“There are no excuses from me at all. I take the responsibility for a very poor second half.”
City’s woes were compounded by the news that Billy Clarke could face up to three months out with a knee injury.