For the Bully Wee a victory would cut the Fifers five points adrift of them and allow some breathing space.
Ultimately it was the hosts who got it their way, Darren Young’s side letting a goal lead slip through their fingers as Clyde hit three in the closing stages of the game.
Clyde started well and with just a couple of minutes gone Jude Smith did well to beat away David Goodwillie’s acrobatic effort.
East Fife paid attention to the early warning and settled into the game nicely.
They enjoyed the bulk of possession during the early stages but too often wasted the final ball.
With so much at stake for both sides, the cagey start to the game was understandable.
Apart from the early chance for Goodwillie, neither keeper was seriously troubled for the next 20 minutes.
East Fife’s best chance of the opening 45 minutes came ten minutes before the break when Scott Mercer broke into the Clyde box.
His cross picked out Kyle Connell but, before he could get his shot away, the Clyde defence were back to clear.
The Fifers ended the first half strongly, giving cause for optimism going into the second 45.
But it was the Bully Wee who created the first opening of the half, Goodwillie’s good work inside the box allowing him to pick out Ross Cunningham.
Cunningham had time and space but fired his effort high over the bar.
East Fife were worthy of their lead when it arrived, though.
Aaron Dunsmore beat his man and got a shot away which Parry did well to get down and block.
First to react to it was Connell who had a simple tap in.
Clyde reacted well to going behind and didn’t take long to get themselves back in the game.
Goodwillie broke clear in behind the Fife defence and sent a terrific chip over Smith.
It got even better for the home side within a matter of minutes when Cunningham put them ahead from a couple of yards out.
East Fife were all of a sudden chasing the game and pressing for an equaliser themselves.
Connell went close as the visitors tried to get level.
But by committing so many men up front they left gaps at the back and were exposed when Cunningham raced clear to make it 3-1.
Fife boss Young said: “We stepped off and, if you give teams that opportunity with that quality of player, then they’ll cause you problems and David Goodwillie does what he does.
"It was his first and maybe only opportunity in the whole game, I thought we defended very well.
"Then someone gets in behind for the second and we don’t match the runner.
"He’s allowed to put the cross in too easy.
"The third one is a counter attack.
"There are large spells of the game where we do well but, at this level, with the quality of the teams, you have to be doing it for 90 minutes.
"There was a small window, five minutes, and we were punished.
"We had opportunities after that but we didn’t take them.”