Parker was taking his first match in charge after replacing Claudio Ranieri as Fulham manager on Thursday. Meanwhile Chelsea boss Maurizio Sarri opted to restore Kepa Arrizabalaga in goal, having dropped the Spain international for the win over Tottenham following his refusal to be substituted during the Carabao Cup final.
But Kepa almost proved the decision to be a mistake when he failed to deal with a routine cross on 18 minutes and, had Fulham forward Ryan Babel been more alert, the Netherlands international would surely have put his side 1-0 ahead.
The goal did not deflate the hosts, however, and only a good save by Kepa prevented Aleksandar Mitrovic from pulling them level, before Calum Chambers did equalise on 28 minutes -- thanks in large part to some shoddy Chelsea defending.
Babel's corner evaded everyone and Chambers, left in acres of space at the back post, rifled the ball into the back of the net before the players ran to celebrate with their new boss.
However, they were level for just three minutes as the much-maligned Jorginho curled in a beautiful effort from just outside the box for his first Chelsea goal since the opening day of the season, having been set up by Eden Hazard.
Mitrovic almost equalised for Fulham in the 88th-minute but forced Kepa into a decent save before Ryan Sessegnon thought he had snatched a point in added time, only to see his effort ruled out for a marginal offside.
Victory brings Chelsea to within two points of fourth-place Manchester United with a game in hand, while Fulham's survival hopes look slimmer and slimmer as they stay 10 points from safety with nine games remaining.