Slater added: "The team has not won in the league since November and recent performances have simply not been good enough.
"Last night's defeat convinced us as a board that change is required now we are still in the hunt for promotion, and that we must appoint a new manager to give us every chance of going up."
The Addicks are fifth in League One, three points off the automatic promotion places.
Parkinson previously had spells as manager of Colchester United and Hull City.
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