Parker was widely expected to leave the Hammers this summer after starring in an otherwise dismal 2010/11 campaign that culminated in relegation from the Premier League.
A number of clubs were thought to be interested in keeping the England midfielder in the top flight, but nobody has yet stepped up their pursuit.
Asked if there had been any bids for Parker, Allardyce told Sky Sports News: "None at all."
Goalkeeper Green is another player to have been linked with a move away from Upton Park and West Brom identified him as a possible target after losing Scott Carson to Bursaspor.
Allardyce admits an offer was turned down but he does not think a transfer will now materialise as he understands the Baggies could be about to sign Ben Foster from Birmingham.
Questioned on whether there had been any bids for the keeper, Allardyce replied: "I think we might have had one from West Brom but that was rejected."
Asked if the two clubs might renegotiate, he added: "No, because I think they have taken Ben Foster haven't they?"
While Parker and Green have been tipped to depart West Ham before the transfer window closes, Allardyce is keen to continue strengthening his squad.
Reading striker Shane Long is thought to be on Allardyce's radar but the manager refused to confirm that an approach had been made, insisting that he wanted to conduct his business professionally.
"I'm not quite sure in that department. I will speak to the chairman tomorrow and see where we are," said Allardyce about Long after Tuesday's friendly against Dagenham & Redbridge.
"We would always like a player that is going to score goals and complement the team, and a player of Shane Long's type, without trying to suggest I am trying to poach him off Reading, because even an interview can get you into trouble today by saying if you want a particular player.
"I don't know if he is available. I will find out off the chairman tomorrow. "