The first half saw few real chances for either team until shortly before half time when Tommy Mooney's shot hit the post with a 20 yard strike. Sammy Igoe had also come close to scoring for Swindon, but he couldn't get his shot on target.
In the second half Swindon hit the poast again, with another shot by Mooney, then Parkin's shot from the rebound was tipped over the car for a Swindon corner.
With their first shot on goal all game Brighton took the lead on 72 minutes Richard Carpenter shot was deflected into the back of the net.
Swindon had more chances to get an equaliser, but Brighton's strong defence held out to give the away side a 1-0 win.
Overall Swindon should have scored at least two goals, but the luck was with Brighton who only managed one on traget shot all game.
Swindon travel to Brighton on Thursday evening for the second leg which they need to win by at least two goals to stop the game going to extra time. The game is also going to be shown live on Sky TV for those not able to make it down to the south coast.