Parliamentary systems don't have that requirement...however, they already have other accountability levers to pull that allows for quick replacement of key players.
Perhaps one of our Aussie, Brit, or Canadian friends can weigh in and fact check me on this, but my understanding of how that works is that there are methods for replacing both the PM (and their Ministers) as well as Parliament.
On the executive branch
1) A no confidence vote can happen, and the Prime Minister (and their Ministers) are ousted and replaced with a new Government.
On the legislative branch
2) A no confidence vote can trigger a general election, which means that people basically have to "re-run" for their MP seats even if their term isn't up yet.
If memory serves, Thatcher came to power in the UK via such measures in 79. A no confidence vote passed by 1 vote, and the end result was a general election getting triggered early, and over 50 labour party seats were lost to members of the conservative party giving them a majority, and making their party leader, Margaret Thatcher, the new PM.