Brazilian midfielder Stefani Miglioranzi turned from villain to hero in a matter of seconds to give Swindon a much-needed win.
The former Portsmouth player should have restored Town's lead in the second half but crashed a shot from close-range against the underside of the crossbar.
Miglioranzi made amends from the corner which followed on 66 minutes. It was flicked on by Tommy Mooney and he was on hand at the far post to secure only Swindon's second victory in the last seven home games with a shot into the roof of the net.
Parkin had given Swindon the lead on the stroke of half time with his 13th of the season. Miglioranzi headed on a free kick from Andy Gurney for Parkin to pick his spot from ten yards.
Parkin's former Chelsea team-mate Leon Knight was left on the bench by Brighton after missing the Boxing Day win against Wycombe through suspension.
Knight was brought on for the second half in place of Chris McPhee and made an immediate impact.
He shot just wide with his first touch and had another effort ruled out by a narrow offside decision before levelling on 55 minutes.
Danny Cullip's free-kick was headed down by Guy Butters for Knight to roll in his 18th of the season.
Brighton's frustration at another narrow away defeat was compounded by a straight red card in stoppage time for Welsh midfielder Nathan Jones for a tackle on Matt Hewlett.
Swindon boss Andy King said: "We are Jeckyl and Hyde. I have spoken in recent weeks about why we cannot get any consistency.
"I think we totally out-played a very good Brighton side and what I have got to find out now is why." Brighton Manager Mark McGhee said: "I was very disappointed with the goals we conceded, particularly the first one. We switched off and that typified the way we defended in the first half."