Parker struck after just six minutes and Rodallega doubled the lead shortly afterwards as the Cottagers came racing out of the traps.
And Rodallega wrapped up the points with a minute to go to surely secure the permanent manager’s job for Kit Symons.
Former Fulham defender, scout and youth coach Symons has transformed the side from the mess he inherited from Felix Magath, and has now won five of his eight matches in charge.
The Cottagers had previously taken just one point from seven under Magath, but they are now out of the drop zone for the first time this season and up to the dizzying heights of 19th.
The players want Symons in charge, as do the fans, but chairman Shahid Khan still seems in no hurry to make the appointment.
However, Khan is due back in the country over the next fortnight when he could finally take what appears the easiest of decisions.
Fulham tore apart a Charlton side who had previously lost just once all season in a breathtaking opening quarter of an hour.
First Parker, who spent six successful years at Charlton at the start of his career, intercepted Chris Solly’s wayward pass and raced forward.
The former England midfielder attempted to tee up Ross McCormack in the area, and although the striker was forced wide he managed to cut the ball back to Parker, who had continued his run and smashed the ball in off the underside of the crossbar.
Six minutes later rampant Fulham doubled their lead when McCormack’s half-volley was superbly saved by Stephen Henderson, only for Rodallega to react first and tuck away the rebound.
A win would have taken Charlton to within a point of leaders Derby but they looked anything but promotion contenders – the closest they came to replying in the first half was when Tim Hoogland’s sliced clearance landed on top of his own crossbar.
Hoogland came agonisingly close to scoring at the right end when his header flew wide and in stoppage time Rodallega volleyed over.
Charlton improved after the break but could not find a way back into the match, with Marcus Bettinelli saving well from George Tucudean’s header and Franck Moussa firing wide.
Instead Fulham wrapped up the victory, and sent an emphatic message to Khan, when Lasse Vigen Christensen set up Rodallega for the third.