Oldham went into the Lancashire/Yorkshire clash having won their four previous matches but they couldn't extend that run any further against Parkinson's men.
With over 2,700 City fans at Boundary Park, first half goals from Nahki Wells and Oldham's Jonson Clarke-Harris saw both sides get a share of the spoils.
Wells opened the scoring for City with his fourteenth goal of the season before Clarke-Harris equalised for the hosts less than three minutes later.
The draw extends City's own unbeaten league run to five games and move them into eighth place in the League 1 table - two points off the Play-Off places.
The Bantams boss was pleased with his side's efforts in the game, with his only real disappointment coming from the fact that City couldn't hang onto their lead for longer.
Parkinson, who celebrated his 46th birthday this Sunday, said: "I was hoping to celebrate it with the three points but we've got the one.
"It was a tough game today. Oldham are a good side who are riding high in terms of confidence given their recent results.
"We knew it would be tough here but we came for all three points and we got a good start in the game with Nahki's goal.
"I just felt after the goal we had a little bit of lapse in concentration though, and it's cost us.
"In the second half, with the way the lads have dug in and really rolled their sleeves up, I thought they were excellent.
"I think we certainly deserved a point from the match.
"We showed great heart in the match. We competed and blocked things. I was very pleased with that side of the game from the lads."
Although a draw was probably a fair reflection on the contest, City could have taken victory from the game if one of their three penalty shouts hadn't been ignored.
Kyel Reid twice went down under challenges in the first hour, with the winger harshly getting booked for diving during the second incident, before Wells appeared to be caught by the out-stretched leg of Jonathan Grounds with time running out.
Parkinson said: "We've had three penalty appeals here today (Sunday) that I will need to have a look at again.
"The two that (Kyel) Reidy was involved in though, he's convinced he has been brought down on both occasions.
"We don't want Kyel getting a reputation that he dives because he doesn't. We tell him not to and he doesn't need to. Kyel has had four big penalty appeals turned down this season and been booked twice for diving as a result. I think he has been harshly treated.
"Looking at Nahki's one (the third penalty shout), I thought that was a penalty myself.
"They probably needed brave calls to be given and it would have been nice to have seen that bravery from the officials."
Following the Oldham draw - City's fifth in their last seven league fixtures - Parkinson's side now have a thirteen-day wait until their next match at home to table-toppers Leyton Orient due to next weekend's F.A Cup with Budweiser Second Round ties.
Parkinson added: "Obviously we are keen to turn some of these draws into wins - we know that - but there certainly isn't a great deal wrong with the way we are playing at the moment.
"This bit of a break gives us a chance to get a few players sorted.
"Garry Thompson, for instance, is going to have an injection in his Achilles. He has been playing with a bit of a problem there for a few games.
"This break gives him a chance to get that right.
"Not just for Garry though, we can use this break for the benefit of the whole squad and come back to what will be a terrific home game (vs Leyton Orient) in good spirits."