The 52-year-old joined the Bees for a fifth spell just eight weeks ago and guided them to a haul of 16 points from eight matches.
However, that did not prove enough to keep them in League Two on the final day of the season.
"A short while ago I had a meeting with the chairman to let him know that I would not be continuing to manage his and your club," said Allen in a letter to supporters, published on the club's website.
"It's been an amazing, unforgettable eight weeks with all the players and the positive energy and the enthusiasm from all the supporters has been quite overwhelming and I can't thank you enough.
"The chairman has offered me a fantastic contract, offered to increase the budget and had done everything he could and no blame what so ever can be put his way. For me, it was eight weeks and I am happy but of course underneath I'm bitterly disappointed and still very upset that we were unable to retain our position in the Football League.
"I really do hope Barnet Football Club can now grow with these players plus the addition of Johnson, Tarpey and Vilhete and you can all be proud to be Barnet supporters."
Allen took over Barnet following the departure of Graham Westley in March.