Parker scored either side of half-time, to notch up his fifth and sixth Premiership goals and steered the Amakhosi to a comfortable win, which puts them back up among the contenders in the league.
Erick Mathoho and Kingston Nkhatha rounded off the result, to lift Chiefs into third on the table, level on points with Moroka Swallows and SuperSport United on 18.
Chiefs wasted little time in gaining control of the encounter, as Mathoho scored the opener within five minutes of the whistle.
Chiefs were awarded a free-kick from a dangerous position, and after some enterprising play on the edge of the area, Mathoho found himself one-on-one with the goalkeeper and blasted the opener past a helpless Modiri Marumo.
City replied with a free-kick of their own, but Mothusi Gopane could only find the safe hands of Itumeleng Khune in goal.
Heavy lightning forced the game off the field 13 minutes into the game, as the hazardous conditions made play impossible.
Having shepherded the players from the field, a 45 minute interval followed, before play resumed as the lightning and heavy rain departed.
Chiefs had a golden opportunity to double their lead in the 23rd minute. Kingston Nkhatha, had time and space to pick his spot, drove his shot straight at Marumo who palmed away Nkhatha’s powerful effort.
But the Zimbabwean international would not be denied a second time, as he burst his way through some weak defending and slotted home having sent Marumo the other way from dead in front.
Chiefs were in complete control and fired at will, with Keegan Ritchie and Nkhatha both forcing Marumo into vital saves in quick succession.
With chances coming thick and fast, Chiefs added their third just six minutes later.
George Lebese led an incisive breakaway along the left flank, before looking up and spotting Parker in the middle.
Lebese centred to Parker, who met the cross with a diving header that sailed into the top left of goal from some way out.
Chiefs headed out into the second period brimming with confidence, but were made to sweat over the fitness of Khune and Nkhatha.
Khune was replaced at half-time by Reyaad Pieterse, while Nkhatha was forced off the field just 10 minutes into the half through injury.
Parker looked unaffected, however, and bagged his second of the night just before the hour mark.
The Bafana Bafana international picked up a loose pass outside the City area, shifted the ball onto his left boot and unleashed a thunderous effort which never looked in doubt of beating Marumo, who tried, but failed to stop the superb long-range missile.
Chiefs were made to work for the last 10 minutes, as Josta Dladla received his marching orders for dissent, allowing City to capitalise.
A lapse in concentration at the back for Chiefs put Puleng Tlolane through on goal, and the second-half replacement made no mistake, as he put away an easy chance, which proved nothing more than a consolation.