But Kevin Lisbie's close range finish with 10 minutes left levelled the score, and Parker completed a remarkable fightback with a fine run and shot in time added on.
Venables's position as Leeds United manager looked increasingly precarious after his team stumbled to a fifth consecutive home defeat in the English premier league.
Charlton Athletic's victory at Elland Road heaped more pressure on the former England coach, whose team have slumped to 16th in the table, only two places above the relegation zone.
Venables immediately faced the television cameras after the match but would not be drawn on his job prospects.
Before the match the former Tottenham Hotspur, Barcelona and England boss had indicated that he would not walk away from the post he took on in July.
"I came to Leeds United because I see it as a club that gives me the chance of winning trophies. That was always the aim and it remains the aim," Venables wrote in the match program.
"Of course the results in recent weeks have not been what any of us wanted. But I'm not throwing in the towel because I don't believe we are in a long-term decline.
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