Draws for Wolves and Middlesbrough gave Birmingham the chance to edge ahead at the top of the table on goal difference, but they rode their luck against the Swans.
Ross Stewart had an early penalty for the visitors saved by Ryan Allsop before Marko Stamenic was sent off after just 26 minutes.
Jay Stansfield opened the scoring just three minutes later but the Blues failed to make the most of a man advantage for over an hour.
A Christoph Klarer own goal brought Swansea level midway through the second half.
But Stansfield's second of the game three minutes from time took Birmingham - co-owned by NFL legend Tom Brady - to the top of the table.
Parker's arrival as West Ham boss this week has done little to revive their fortunes.
The Hammers came from 3-0 down to snatch a 3-3 draw in Parker's first game in charge at Cardiff on Wednesday, but failed to repeat the trick after falling two behind at Blackburn.
Andri Gudjohnsen and Dapo Afolayan put Rovers in front before Joel Piroe pulled a consolation goal back.
West Ham slip out of the play-off places down to ninth, six points behind Birmingham.
Cardiff produced a remarkable comeback from 4-1 down to beat Burnley 5-4 in the game of the day.
Alex Robertson made amends for an earlier own goal when he struck deep into stoppage time to complete the fightback.
Defeat keeps Burnley, who were relegated from the Premier League last season, in the bottom three.
Stoke came from behind to move level on points with Birmingham and Middlesbrough with a 2-1 win at Portsmouth.
Bottom-of-the-table Preston shocked Southampton 1-0 with Jordan Thompson's goal giving Graham Alexander a first win since taking charge at the end of September.
Wolves missed the chance to go top earlier on Saturday when they blew the lead in a 1-1 draw at Wrexham.
Middlesbrough again missed the presence of the injured Will Lankshear as they failed to break down struggling Watford in a 0-0 draw at Vicarage Road.
