The Potters were in negotiations with the Hammers for much of the week leading up to last Wednesday's transfer window deadline but missed out because of the player's wish to remain in London.
Parker eventually signed for Tottenham, with Pulis instead signing Spurs midfielder Wilson Palacios in addition to striker Peter Crouch and Birmingham's Cameron Jerome.
"We did have a go for Parker. It was something we were trying to do for a few weeks," said the Stoke boss in The Sentinel.
"But we realised that if he could, he wanted to stay in London. That was a big part of it with him. I felt there were certain areas of the team we can improve and have to improve and that was one of them.
"I enquired about him and kept in touch with Sam Allardyce (West Ham manager) to find out what the situation was. But Sam said Parker would prefer to stay in London because he'd got three boys in school there."