The defender was a player-coach at Park View Road last term as the club just missed out on the National League South play-offs.
But the team are under new management following Steve King’s appointment.
Parkinson, the former Woking and Dover centre-back, wanted to continue coaching as well as playing but an agreement couldn’t be reached.
King told the club’s website: “Unfortunately I couldn’t offer Jack the coaching role he wanted as I will be bringing my own team in.
“I would have loved to kept Jack at the club in a playing capacity as he is a top player, but we completely understand his decision and wish him well for the future and thank him for all he has done for Welling.”
King has already started shaping his squad, with four new signings already announced.