Allardyce's men went ahead inside the first four minutes at Upton Park thanks a Joe Cole strike, only for West Brom striker Nicolas Anelka to turn things around with a first-half double.
West Ham recovered excellently in the second half and appeared to be on their way to victory as goals from Modibo Maiga and Kevin Nolan put them back in front.
However, the hosts could not hang on, with Saido Berahino's 69th minute strike ensuring a point for Keith Downing's men, leaving Allardyce - who saw defender James Tomkins limp off after less than 10 minutes - to bemoan West Ham's failure to maintain a lead.
"The game was very frustrating," Allardyce told BBC Sport. "There is a reason why that happened, none of our centre-backs are fit.
"Losing James Tomkins was a big blow for us. Our coping mechanism was playing four full-backs across the back four. That has cost us three points.
"They did their very best but got caught out. We didn't have enough aerial threat defending corners."
Allardyce did credit his team for the character displayed in fighting back from 2-1 down, but was quick to criticise their lack of defensive concentration.
"We showed great fighting spirit in the second half," he added.
"We practice on the training ground how to hold onto leads. But the players forgot how to deal with West Brom's kick-off and stop them scoring.
"You can blame anxiety or fatigue, but you can't forget how to deal with them."
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