The game looked to be heading for a draw until former Chelsea and Charlton midfielder Parker weaved his way through the Boro defence before shooting past Mark Schwarzer and into the far corner of the net for a brilliant goal.
The defeat was a blow to Middlesbrough following back to back wins over Arsenal and Derby County but a valuable three points for the London side.
After the final whistle jubilant West Ham players threw their shirts to fans who had made the 500-mile round trip to the north-east on a bitterly cold day.
Ironically, Middlesbrough began the game full of confidence after picking up seven points from their previous three games. They were full of running and enterprising in their attacking play.
Turkey international Tuncay was hoping to boost his tally after scoring in each of those three matches.
The game was played at a fast pace but ironically neither goalkeeper was tested in the first half.
Middlesbrough skipper George Boateng was booked for a 31st minute foul, a harsh decision that did not go down well with the home fans.
Two minutes later Parker was also cautioned for a foul bit Boro took the lead in the 40th minute when local product David Wheater scored from close range.
The goal came from a Stewart Downing free-kick following a foul by Jonathan Spector on Jeremy Aliadiere.
Aliadiere was on the spot to connect with the free-kick and head into the six-yard area where Wheater connected.
Middlesbrough deserved the goal but they were stunned just before the break when West Ham drew level with a superb Dean Ashton volley.
The former Crewe and Norwich striker hit a shot of tremendous power and accuracy into the corner of the net, leaving Schwarzer with no chance of stopping it.
The goal boosted West Ham's confidence and they began the second half in an attacking mood. Nolberto Solano saw his brilliant free-kick tipped over the bar but a minute later was booked for time-wasting.
The Hammers continued to press strongly and the Middlesbrough defence had their work cut out to hold them.
Boro strove hard without showing the cutting edge for a goal before Parker showed them how it should be done.
Middlesbrough manager Gareth Southgate now has to pick his players up for a Boxing Day visit to relegation rivals Birmingham City.