City shrugged off the early blow of seeing Alan Sheehan's penalty saved to take a tenth minute lead via an Louis Thompson own goal.
The lead was a deserved one after an impressive start to the game from Parkinson's men but two goals from Jonathan Obika - both from corners - eventually proved enough to see Swindon finish the game with the points.
Parkinson said: "We worked so hard this week to prepare for this game because we were determined to put in a better performance than we did against Yeovil Town last Saturday.
"I thought we started the game tremendously well. Got the penalty, missed it, went again, and then scored a good goal thanks to great play from Billy Clarke and (Mark) Yeatesy and JK (Jason Kennedy) driving into the box.
"Our general play, in fact, throughout the game was good.
"We can't move away, however, from the fact that we have conceded five goals in our last two home games and four of them have come from set-pieces. That is not good enough.
"We can be encouraged by the fact that we did play better today (Saturday) than we did against Yeovil but to give the opposition such a lift by the goals we have conceded in the last two home games is so unlike us.
"I certainly can't remember a time as manager here where such a high percentage of our goals conceded have come from those situations. It is very frustrating.
"We clearly need to get better with that.
"We have started so well today and that is why I am so annoyed after the game because we have given Swindon a huge lift by the goals we have scored.
"Our passing and general play was much better today than last weekend but goals change everything.
"It was a better performance by some distance this Saturday from last time out but we have still ended up with nothing.
"To get beaten at home again is not good enough so we have to make sure we are ready for our next match, which is MK Dons away on Tuesday.
"We must do better as a group, and we will. We will be looking to go again at MK Dons."