Okonkwo first failed to collect Ethan Galbraith's free kick as his punch looped into the air behind him and towards Ben Cabango.
- Reynolds, Mac sell minority stake in Wrexham
The Swansea captain sent a looping header towards goal, but Okonkwo was back in position to make an easy catch by the post. But he inexplicably dropped the ball and Adam Idah had the simple task of nudging it across the line.
"If you look at the moment the punch hasn't gone where he wanted it to go," boss Parkinson said after Wrexham's winless Championship run was extended to five matches.
"He's tried to rectify his mistake and I think Dom [Hyam] could have headed the ball out, but he's kind of got over the top of him and they've almost got in each other's way.
"It's a high-profile mistake but Arthur's been outstanding. He's been a colossus of a goalkeeper, and he'll get every bit of support because he's won us a lot of points over the last few years.
"When you make a mistake like that it's going to be highlighted, but he's a tough character and he'll get great support from his teammates."
Wrexham's misery was compounded by a serious injury to midfielder Lewis O'Brien, who had only been on the pitch for seven minutes before he was scythed down by Galbraith.
Parkinson said: "He [Galbraith] is lucky not to get red. Lewis has injured his knee and dislocated his shoulder.
"He's a tough player and he's not coming off unless it's something really serious. It's a tough way to lose a game of football."
Idah's winner came after Zan Vipotnik had equalised for Swansea after 70 minutes with a deflected shot off Hyam.
Wrexham had led from the 14th minute when Cameron Burgess headed Ryan Longman's cross into his own net, but Swansea were thereafter the better team without fashioning clear-cut chances.
"It was a really emotional game like we all expected," said Swansea manager Vitor Matos, who has now overseen three successive home wins to lift his side to 17th place and within two points of Wrexham.
"They started the game well, but the reaction after the goal was excellent. I think we got the reward of that.
"Adam is a striker with a hunger to score. He needs that for himself because he hasn't been starting and wants to make an impact when he comes on.
"The idea was to bring him on and let him do that so I'm really happy for him.
"At this moment every point helps, that's the important thing. Of course you try to get up [the table], but there's so many games to come."
