The Hollywood-backed Welsh side, co-owned by actors Ryan Reynolds and Rob McElhenney, are aiming for a fourth consecutive promotion to continue their remarkable rise through the English football pyramid.
Victory would have taken Wrexham into the play-off places and they twice led through Ollie Rathbone and a Joe Williams own goal.
Sinclair Armstrong got the home side's first equaliser just after half-time and Bird's fine effort on 89 minutes delivered a vital point in their own push for the play-offs.
Wrexham move level on points with sixth-placed Derby but outside the play-off places on goal difference, with City just one point further back in ninth.
Portsmouth boosted their chances of survival with a 3-1 win at fellow strugglers Charlton.
Victory takes Pompey four points clear of the relegation zone in another blow to Leicester's hopes of survival.
The Foxes sit in the bottom three after being deducted six points for breaching financial rules and are in danger of sinking to the third tier just a decade on from winning the Premier League.