The home side went behind early, as Mozambique's Diogo Alberto silenced the Cape Town Stadium with a soft strike which goalkeeper Itumeleng Khune failed to keep out.
But shortly before the interval, Parker drew the teams level from the penalty spot, before Kekana scored a long-range stinger, to help South Africa toward all three points.
Parker sealed his brace seven minutes from the end, as the Mambas were brushed aside, sending South Africa on their way to the perfect start in their Chan campaign.
He carved out an early opportunity for the hosts when Lerato Chabangu stole possession in a dangerous position, but goalkeeper Soares Soares made a simple stop to deny the some side.
Mozambique, however, were in the mood to spoil South Africa's party, as Diogo's tame effort caught skipper Itumeleng Khune off guard, with the Kaizer Chiefs goalkeeper slipping on the wet turf in the process, and handing the visitors the early advantage as the ball bobbled over the goalline.
Bafana came back strongly and before long, Sibusiso Vilakazi forced a superb stop from Soarito, after the powerful midfielder rose well and guided his headed effort toward goal.
Mashego grazed the crossbar from a freekick on 25 minutes, before Chabangu sent his effort high of the mark, as South Africa continued to press for the equaliser.
The pressure eventually told and a minute later, Vilakazi was brought down inside the area and was awarded a penalty for his efforts. Parker dispatched it into the right of goal, sending Soarito the other way.
It was all going South Africa's way, save for putting the goals away, as Mashego struck the upright and was denied a superb finish, following some eye-catching interplay in the middle of the park.
Back after half-time and level at a goal apiece, Mashego was in with a good chance after the break, but Soares got himself in the way, as the home side looked to pick up where they left off in the opening period.
Bafana would have to wait a quarter-of-an-hour for their next attempt, but it seemed worth the wait, as Kekana's wonder-goal from 30 yards out stunned the Mambas and handed the South Africans a vital advantage.
Kekana picked up a pass from debutant Lindokuhle Mbatha, trotted ahead and unleashed a contender for Goal-of-the-tournament as the ball sailed into the back of the net.
The hosts' second brought the home supporters to life and momentum stayed with the South Africans, who looked deadly in attack.
Parker wrapped-up the result in the last 10 minutes when he finished off a solid build-up in midfield, scoring his second of the night and fourth against the Mozambique in his last two outings.