The Cottagers' immediate return to the Championship was confirmed on 2 April but Parker had restored pride in recent weeks, overseeing a run of three wins in four games before the Magpies' visit to Craven Cottage.
The woeful defending that has characterised their season was on full display again, though, as Jonjo Shelvey and Ayoze Perez scored within the space of just over two first-half minutes to put the visitors ahead before Fabian Schar netted a third in the second half and Salomon Rondon added late gloss to a comprehensive scoreline.
Parker will now turn his attention to masterminding a swift return to the Premier League, while Newcastle manager Benitez will continue his discussions with owner Mike Ashley – who was watching on in west London – over a new contract.
Newcastle wasted little time in forging ahead, Shelvey taking down Matt Ritchie's corner 18 yards out and unleashing a superb half-volley into Sergio Rico's top-right corner.
Things got worse for the hosts just 124 seconds later when Rico fumbled Christian Atsu's low effort into the path of Perez, who prodded home from close range for his sixth goal in his last five Premier League games.
Aleksandar Mitrovic headed narrowly wide from a corner midway through the first half as Fulham belatedly woke from their slumber, but they were unable to reduce the deficit before the interval.
Fulham improved at the start of the second period but they were undone by more generous defending when Schar easily out-jumped Alfie Mawson to head home Ritchie's corner in the 61st minute.
The host team's misery was compounded in the 90th minute when Rondon powered through and slotted in from an acute angle after Rico had saved his initial effort.