Parker was confirmed as the successor to Nuno Espirito Santo in the hours before the trip to Wales, but was stunned by a first-half Jack Moylan hat-trick for the Bluebirds.
The Irish international blasted home a stunning strike to open the scoring before doubling Cardiff's lead from the penalty spot and then heading in a third from a corner.
Jarrod Bowen helped haul West Ham back into the game as he set up Divine Mukasa before Arne Engels netted his first goal for the club since a big money move from Celtic.
Max Kilman grabbed the equaliser eight minutes from time thanks to another Bowen assist as both sides had to settle for a point.
West Ham remain in eighth but are only five points off the top after Boro lost in the league for the first time since August.
Wrexham, backed by Hollywood actors Ryan Reynolds and Rob McElhenney, have struggled so far this season but climbed to 16th with just their second win in six games.
George Thomason and Ollie Rathbone scored the goals as Middlesbrough badly missed the presence of the league's top goalscorer Will Lankshear due to injury.
Sean Neave struck twice in the final 10 minutes for Wolves to secure a 2-1 win at Bristol City.
The Newcastle loanee's double takes Wolves up to second and within a point of Middlesbrough.