The Addicks have now failed to win in their last 12 Championship matches and have taken a miserly four points from a possible 36.
In the four matches Parkinson has been in charge they've managed just one point out of 12 and unless action is taken swiftly to get a permanent manager in, the side which is totally lacking in confidence look a good bet for relegation even at this early stage in the season.
Coventry were nothing more than workmanlike and in a dreadful first half it took over half an hour for either goalkeeper to have a shot to save.
Charlton's woes started as early as the 13th minute when on loan defender Jay McEveley fell heavily and went off with a shoulder injury.
The home side's only meaningful chance of the first half was a header from skipper Mark Hudson that was comfortably saved.
The Sky Blues took the lead five minutes before half time when the Charlton defence failed to clear Daniel Fox's left-wing cross and Robbie Simpson fired the ball home.
Charlton had a chance to get themselves back in the match three minutes after the break when they were awarded a dubious penalty.
Ben Turner was adjudged to have brought down Izale McLeod and Charlton's on-loan striker Deon Burton scored from the spot, his penalty going in off the underside of the crossbar.
But within two minutes Coventry had regained the lead when a 30-yard free-kick from Fox eluded everybody and went into the Charlton goal via the crossbar.
Charlton put on two more substitutes in an effort to get a second equaliser, but Coventry comfortably held out for a useful away victory.