While Laidlaw seemed to kick everything that moved to drag his side back into the game after a shaky start, Parks suffered a difficult evening as he missed several kicks at goal before being replaced just before the hour.
Alan Solomons' Edinburgh has won their last three home games and it was easy to see why in the opening minutes as they started like an express train and took a third minute lead thanks to a Laidlaw penalty.
Parks responded for Connacht, who have not won away in the tournament since April, with a drop goal on 11 minutes as the visitors battled their way back into the game.
And it got better for Connacht on 15 minutes as Robbie Henshaw went over for a score, with Parks adding the extras - at this stage the most unlikely of wins seemed on the cards.
Laidlaw knocked over penalties on 20 and 29 minutes to get within a point of the visitors as Edinburgh started to boss both territory and possession.
Kicking seemed as though it would play a crucial role in proceedings if Connacht were to escape Murrayfield with a win but on 35 minutes Parks missed a crucial penalty to extend his side's lead.
The kick was to prove decisive as Dougie Fife scored Edinburgh's first try of the evening on the stroke of half-time, with Laidlaw adding the extras to swing the game in the home sides favour 16-10 at the break.
Connacht looked dead and buried ten minutes into the second-half as Laidlaw extended Edinburgh's lead to 19-10 with a penalty.
Parks was experiencing a mixed evening in a Connacht shirt and on 56 minutes he missed a penalty to drag his side back into the game and just 60 seconds later he was replaced by Craig Ronaldson.
And just two minutes later Parks' miss looked even worse as Laidlaw slotted a penalty of his own to put Edinburgh in a commanding position at 22-10 with just 20 minutes to play.
Cornell du Preez ensured there was to be no fightback for Connacht on 73 minutes as he went over for an Edinburgh score - inevitably the ever-reliable Laidlaw converted the try to make the score 29-10.
Down and out; Connacht collapsed in the final stages as Nick de Luca went over for Edinburgh's third try of the evening on 77 minutes - Laidlaw adding the extras.
And Edinburgh confirmed a bonus point as WP Nel scored two minutes later - again Laidlaw adding the extras - as the home side claimed a 43-10 win.
|Edinburgh Rugby Score Card|
|Nick de Luca||1||5|
|Cornell du Preez||1||5|
|Connacht Rugby Score Card|