Coventry were heading for a 3-2 win when defender Jordan Clarke handled as he challenged striker James Hanson just inside the visitors' box, referee Kevin Friend pointed to the spot and up stepped Nahki Wells to score to complete his hat-trick in the fourth minute of stoppage time.
The Sky Blues boss accused the Bantams of playing "football from the dark ages" but Parkinson, who praised the spirit of his players, said of Pressley, a former Scottish international defender who played with Dundee United, Hearts and Celtic: "He has only just arrived in England so he isn't aware of our progress over the last 18 months.
"We play with balance. We like to get the ball out wide and deliver crosses into the box. The truth is that it is exciting to watch."
Angry words were exchanged between the rival managers on occasions and Parkinson admitted: "Emotions were running high. He came into our box to have a go at us, but it was just a few words, nothing more than that."
Bradford, who are without a win in seven league and cup matches, found themselves two goals down in seven minutes, but hit back through Wells to level the scores after 28 minutes only for Coventry to go 3-2 up in the 42nd minute.
Andy Webster scored the first in the second minute following a free-kick from skipper Carl Baker. Keeper Jon McLaughlin tried to catch the ball, but he appeared to be impeded by team-mate Rory McArdle and the ball ran loose to Webster to fire the ball into the net.
Before Bradford recovered, Callum Wilson took the ball off McArdle after chasing down a long pass from Blair Adams and, although McLaughlin knocked the ball away from him, it ran loose to fellow striker Leon Clarke to fire into the unguarded net for his 12th goal of the season.
Hanson saw his far-post header stopped on the line by keeper Joe Murphy before Wells pulled a goal back after 18 minutes when Hanson headed a long clearance from Kyel Reid over Murphy's head and into the path of Wells.
Hanson and Wells combined again for Bradford's equaliser, Hanson heading James Meredith's free-kick into the path of the Bermudian international to volley into the net.
However, Wilson restored Coventry's lead three minutes before half-time with a close range angled shot after getting the better of defender Matthew Bates.
Both sides had chances in the second half before Wells' last-gasp equaliser.
Parkinson said: "It was a great game, credit to both teams. We had worked all week to get rid ourselves of the poor performance at Rotherham in the FA Cup only to find ourselves 2-0 down so quickly and leave ourselves a lot of catching up to do. But we showed a lot of character and I thought we played terrifically well, when you take the circumstances into account.
"Individual mistakes cost us at the beginning and that was disappointing from our point of view.
"Steven Pressley must be delighted that he has inherited these two strikers (Wilson and Clarke). They are good players who score goals and when you have two players like that it takes the pressure off the rest of the side.
"But we also caused them problems, using our two wide players and I don't think Coventry could cope with them."
Parkinson praised hat-trick hero Wells, who has just returned to the side after injury. He said: "Sometimes you miss football and get rustiness in your game, but he was superb."
The manager admitted he didn't watch Wells' penalty.
"I just waited for the crowd's reaction," he said.