The Australian defender came to the rescue after Leicester had been reduced to ten men by the second-half dismissal of Stephen Hughes.
Reid swivelled to crack in a left-foot drive from 15 yards, his second goal of the season.
It cancelled out David Connolly's 33rd minute penalty for Leicester, the seventh spot-kick awarded against Brighton in their last ten matches and the third in as many games.
Referee Mike Thorpe spotted a handball inside the box by on-loan defender Joe Dolan on his home debut.
It was rough justice on Brighton, who had dominated up to that point and had four penalty appeals of their own rejected by Thorpe.
Target man Mark McCammon went down inside the box under challenges from Nikos Dabizas early on and Leicester keeper Ian Walker on the stroke of half-time.
The Seagulls had two further claims denied after the break, for handball from a McCammon header from a corner and for a tackle by Alan Maybury on substitute Jake Robinson.
Hughes handed Brighton a lifeline with two yellow cards for kicking the ball away.
The first came in the 50th minute and prevented Brighton from taking a free-kick inside the centre circle.
Leicester's Scottish midfielder repeated the offence 12 minutes later after he had been given offside.
Reid's late leveller meant Leicester had to settle for their 19th draw of the season as they stretched their unbeaten run to six matches.