I agree with all that, but that focuses mainly on the Commons, while the somewhat odd (By our standards) institution of the Lords and their admixture in and out of executive offices and policy affairs is a curve ball that makes it very difficult to say exactly where one branch ends and another begins, further complicated by the fact that contrary to American preconceived ideas, the personal politics of the Lords are as likely to be as bizarrely leftist as anything else, a product of the rather unique English educational system I suppose.
Party discipline has certainly made recent UK politics look a lot more like American ones, how long that will last remains to be seen because it doesn't seem to have been the general trend for that type of political organization's long-term behavior. While third parties evaporate like a water drop in a hot frying pan in the US, they seem to have a tenacious persistence about them in countries with a Parliamentary system...In Israel, for instance, the balance of power has been held for most of its existence by tiny splinter parties in the Knesset, often with quite bizarre ideas and agendas of their own, since neither Labour nor Likud could ever muster a clean majority without getting enough of a half-dozen assorted oddball parties to sign on in a coalition with them, and constrained long afterward to the terms of their deals in order to keep the sitting government in office (In the US system of course, with its defined terms of office, junior partners in the alliances necessary to win office are frequently thrown to the wolves as soon as the election results are certified). In Weimar Germany, the Nazis never won a clean majority, but had to engage in coalition politics and a couple of extraordinary circumstances and legal loopholes (Mainly the Emergency Laws and the lack of any prohibition on one man holding two principal offices at once) to be able to seize power when they did.