Afterwards Thordarson admitted: "I was very disappointed with the first half, we didn't start well.
"I could see their goal coming. Our commitment and work rate was not good enough and there were strong words from me at half time.
"We were far better in the second half, deserved our equaliser and should have won and I am not happy with just a point.
"Now I am looking for a performance that lasts 90 minutes and not 45 minutes at home." Swindon boss Andy King said: "We should have won the game after the first half but we lost our way in the second half and they probably deserved to win the game in the last 15 minutes.
"In the second half we played too deep and I was never comfortable. I didn't recognise my team in the second half, we lost our shape and individuals didn't do things right.
"Basically we were pummelled but we came away with a point and remain unbeaten." Swindon took the lead in the 20th minute when young striker Sam Parkin grabbed his fourth goal of the season. He was left in space unchecked when David Duke picked him out with a clever cross from the left. Parkin calmly turned and smashed his shot in from 15 yards.
Swindon keeper Rhys Evans was kept busy as Barnsley looked for an equaliser but a string of fine saves kept the home side at bay.
The breakthrough finally came in the 77th minute when Kevin Betsy cut in from the right and smashed a blistering shot that cannoned off the bar.
Fallon followed up smartly and coolly side-footed his shot past Evans with his third goal of the season to equalise.
In the dying minutes Fallon went close with a header that hit the bar and at the other end Tommy Mooney was also denied by the woodwork as he crashed his shot against the upright.