However, there are many parallels.
One I find intriguing is the eventual requirement to separate the warring forces, as much as possible, into 2 separate nations.
Cromwell did bring a religious element in with his, but we could say that the US is bringing a political element in that is at odds with the religious history of Iraq. The democratization of Iraq is a contrary doctrine to Islam. This democratization is a matter of faith.....that all the world would be better were it democratic.
From the muslim point of view, your Cromwell allusion is probably apt.