Parkin made no mistake when five minutes later he had the ball in the back of the net again from a shot from inside the box, and this time his goal counted, making it his sixth goal of the season so far.
James Milner came close again for Swindon, but couldn't score his first goal of the season. He also won't have a chance on Wednesday night as he is cup tied for Swindon's trip to Leeds in the Carling Cup.
In the second half Barnsley had a lot more of the ball, and it was just a question of when they would score their first after Swindon had much of the ball in the first half.
After many chances Barnsley finally scored their first and only goal of the game to make the score 1-1 on 77 minutes when Betsy hit the bar only for the ball to bounce back to to fall to Fallon to score.
Both teams then had chances to score the winner. Swindon's best chance came with a minute of normal time left on the clock when Tommy Mooney beat the Barnsley keeper, but his shot hit the post from a tight angle.
Overall a point was a fair result for both teams, who had times both controlled the play with Swindon having the better first half, and Barnsley having the better second half.
Swindopn now travel past Barnsley again on Wednesday when they visit Yorkshire again to play Leeds United in the Carling League Cup.