Steven Schumacher was pushed in the back by Craig Ireland, but Symes hit a weak spot-kick wide of the keeper's right-hand post.
In truth it would have been more than the home side deserved in what was a scrappy affair save for the frantic last five minutes.
City started brightly and took the lead through Dean Windass who made it two in two games. Nicky Summerbee did well to beat David Farrell on the right-hand touchline.
He fired a low cross that was smashed home by Windass on the turn.
Although not creating many chances Bradford were in control of the game until another blunder by rookie keeper Ross Turnbull let the visitors back in.
Peter Kennedy's left-wing corner found the head of 6ft 4in striker Clive Platt. Turner looked to have the effort covered but he somehow managed to let the ball slip under his body.
The confidence that Bradford's youngsters seemed to have shown seemed to be lost in that moment and Peterborough came more and more into the game.
Things went from bad to worse when Windass was stretchered off with what looks like a serious ankle injury.
The second half started in mundane fashion but Peterborough took the lead 10 minutes in.
Thirty-eight-year-old Andy Legg's long throw was headed on by Platt and Kennedy hit a sweet volley into the corner of Turnbull's net.
Without Windass on the pitch City lacked any cutting edge to force their way back into the game. Their best chance was a well hit volley by Ben Muirhead from 20 yards.
A disappointing crowd of 6929 became increasingly frustrated at City's inability to create an opening, that was until the late equaliser.
Again it was Muirhead with the drive which was heading well wide until Wetherall deflected it past a stranded Mark Tyler in the Posh goal to set up a frantic finish.
New boss Colin Todd will be happy that his new look side are off the mark but will know they will have to improve on this showing if they are to avoid another relegation battle.