Simpson added: "It is with some sadness that we have taken the decision, along with Aidy, that the time is right for a new manager to take Watford forward.
"We have mutually decided that a parting of the ways is in the best interests of all concerned."
Watford lost in the play-off semi-final to eventual winners Hull last season after leading the table at one stage and have won just 12 out of 48 league matches since last November.
The Hornets take on Swansea at the Liberty Stadium on Saturday but have not confirmed whether first-team coach Martin Hunter will be in charge.
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