Why do people support parliamentary systems?

Because its a system of democratic governance of a state where the executive branch derives its democratic legitimacy from the legislature (parliament) and is also held accountable to that legislature. In a parliamentary system, the head of state is normally a different person from the head of government. This is in contrast to a presidential system in a democracy, where the head of state often is also the head of government, and most importantly, the executive branch does not derive its democratic legitimacy from the legislature.