Cole's backpass led directly to Jermain Defoe's equaliser at Upton Park as Tottenham went on to claim a 2-1 win - but Cole had opened the scoring with a spectacular volley and stretched the visitors' defence as a lone striker.
With Hammers boss Gianfranco Zola short on numbers in attack, Cole's next opportunity is likely to be against Millwall in the Carling Cup.
Parker said: ''Carlton has got all the attributes to be a big player for us and at international level. Everyone could see against Spurs he's got massive strength.
''He can bring people into the game as well and he's got a massive season ahead of him. If he does well this season I think he'll definitely be pushing to get into the England side.
''He scored a great goal and he's made a mistake but that could have happened to any of us out there, it's part and parcel of football. You put yourself on offer like Coley does and you are going to make mistakes.''
Technical director Gianluca Nani and chief executive Scott Duxbury are looking to add at least one more forward before the transfer window closes.
Bordeaux's Marouane Chamakh has ''rejected'' a move but the Hammers are understood still to be working on the deal, while Lazio's Goran Pandev and Livorno's Alessandro Diamanti have been linked in Italy with playing for Zola.
''We would love to bring some more players in,'' Parker added. ''Our squad is very, very short and we are short on numbers. We all understand the situation we are in and it's what we have to work with.
''We have to dig in - and we'll do that and try and get some results. If we play and perform like we did against Spurs we'll be okay.
''There's no hiding the fact that Tottenham are going to be up there this year, they are a very good side and have a very good squad. If we can perform like we did against them there's not going to be many teams who can live with us.''
A heavy police presence is anticipated for the second-round clash against Millwall, West Ham's bitter rivals.
''I don't have a background in the match but they tell me it's a game that they have suggested I don't come to by walking to the stadium,'' Zola joked. ''It's going to be, obviously, very much a heated game. We are ready, we are professional, we care about the competition and we'll try to do our best.''