The Bantams stretched their unbeaten run to five matches. but only one of those games has resulted in a win for the Capital One Cup runners-up and they now lie nine points off the top seven albeit with two games in hand.
Both sides hit the woodwork, with Jason Banton's angled shot smashing off an upright for the hosts before City target man Garry Thompson saw his 80th-minute volley beat home `keeper Jake Cole but loop up off the top of the crossbar.
And Parkinson believes Bradford can still make the play-offs if they show more composure in front of goal.
He said: "I don't think we could have done much more to win the game. It was a bit scrappy at times but that's because there is so much at stake for both teams but I thought we gave everything.
"We couldn't have done much more, we hit the bar, had a one-on-one and two headers in the six-yard box. I can remember as a player going away from home and creating one chance and someone scoring and you win.
"We knew it coming down here, it's a long old trip on a Tuesday night. This is a difficult place to come, Plymouth are a big club and John (Sheridan, Argyle boss) has started to get them going and they are a decent side.
"Top teams, bottom teams there is very little in it, but we are a team at the moment who just need a bit of luck, a bit of inspiration in front of goal to get us going.
"That's five unbeaten since the cup final and we have got to keep doing the right things in the right area and believe and it will happen for us."
- Sports & Recreation