The Blues were on a high after booking their place in the Carling Cup final in midweek but they were brought back down to earth as Marlon King and Richard Wood put the visitors two up with 26 minutes gone.
David Bentley reduced the deficit before half-time with a stunning strike from 25 yards for his first Blues goal and, after a sustained spell of pressure, two goals in six minutes from Parnaby and Phillips midway through the second half completed the turnaround.
Coventry were hoping the cup could provide a change in fortunes after no wins in eight in the league while Blues have also struggled in the Premier League of late.
It was the hosts, showing nine changes from Wednesday's victory over West Ham, who began the brighter and they could have been two up in the opening nine minutes from almost identical chances.
First Bentley broke away down the left before crossing for Nikola Zigic, who should have done better than head wide, then David Murphy produced an even better cross for Phillips. His header was brilliantly tipped onto the crossbar by highly-rated keeper Keiren Westwood.
However, when the breakthrough did come in the 11th minute, it was Colin Doyle who was picking the ball out of his net. King was given time to turn 25 yards out and drove into the box before sending his shot just inside the far post.
The goal lifted the Sky Blues and they piled on the pressure with a series of corners, which paid dividends in the 26th minute as former Blue Gary McSheffrey picked out the head of Wood.
There is less than 20 miles between Coventry and Birmingham, and the visiting fans were revelling in the discomfort of their local rivals.
Alex McLeish's side had plenty of the ball but could not break down a green and black wall, while, in contrast, Coventry were having no trouble finding space in and around the home defence.
The hosts needed a moment of inspiration, and they got it when Bentley drilled in an unstoppable shot from 25 yards with 35 minutes gone.
Westwood was in the right place but was beaten by the sheer pace on the ball, which went in off the underside of the crossbar.
The second half was 10 minutes late kicking off as the officials made some running repairs to Westwood's net with sticky tape, and when the action did finally resume it was the home side who made the stronger start.
A smart turn from Bentley gave the midfielder another glimpse of goal but this time his shot was well wide, while Jean Beausejour should have done much better than flick a header well off target when the impressive on loan from Tottenham man picked him out in a great position.
At the other end, King sent a shot wide of the far post but the pressure on the visitors was growing. And in the 67th minute that brought a first Blues goal for Parnaby, three and a half years after he joined the club from Middlesbrough and nearly five years after he last found the net.
The 28-year-old has found his opportunities increasingly limited at St Andrew's but he was in the right place to hook the ball into an empty net after Alexander Hleb had drawn Westwood to the near post.
Phillips had looked sharp despite a lack of games for Blues this season, and he drew another smart save from Westwood with a fierce near-post effort in the 72nd minute.
There was no denying the 37-year-old moments later, though, as a corner to the far post from Beausejour picked out the striker, who lashed a volley into the far corner.
Phillips, whose only other goal came in the Carling Cup in October, has been told he can leave St Andrew's but he would certainly have sent a message to McLeish with this performance.
Aidy Boothroyd sent on joint leading scorer Freddy Eastwood in a bid to salvage the game but, although Coventry managed a decent spell of late pressure, they could not find the equaliser.
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