No – it’s the other way round, oddly enough. When you have a parliamentary system, in which the legislature chooses the prime minister, there is far more party discipline in the legislature, and the legislature polarises into government versus opposition. Even if there are multiple parties, in the longer term they teld to form stable coalitions.
And if you look at the US Congress, even though there are nominally two parties, in fact both houses are full of people acting pretty independently, with coalitions forming from time to time around various issues. The only time when party discipline becomes important is when electing various office-holders (e.g., the Speaker and the chairs of the committees).
There are whole organisations devoted to tracking the voting records of members of the US Congress. No one (except perhaps the party whips) tracks the voting records of the British House of Commons, because practically all the time they vote rigidly on party lines.
If you changed the voting system for the US Congress to bring in instant run-off voting or proportional representation, you’d find members of the US Congress behaving even more independently than they do now, as well as minor parties splitting off in every direction.