"On Monday Phil called to accept the position and told Charlton," Terriers vice-chairman Andrew Watson said in a club statement.
"We had agreed he would continue in that role until the end of the season or until Charlton's Premiership status was secured.
"On Tuesday he called to reiterate and asked the club to tie up when the announcement would be made with Charlton chief executive Peter Varney.
"Peter Varney called before lunch yesterday and both parties agreed to announce the appointment at 9.30 this morning.
"Phil then called last night to say that Charlton had made him an unbelievable offer to stay at The Valley and he needed until this morning to decide."
Former Colchester United boss Parkinson has said: "I have had a change of heart. I have been enormously impressed with Huddersfield and the integrity of the board. It is a fantastic club with a great infrastructure which will go from strength to strength.
"However, the lure of working with Alan Pardew and in the Premiership was not something I could walk away from."