A "parliamentary system" is not about just having a parliament (an elected legislature), it is a name for a system where the elected parliament is the paramount power in government, including choosing the head of government. The French have something close to that, the US does not.
The French have a semi-presidential system where, as in a parliamentary system, the head of government is the PM elected by the parliament and formally appointed by the President who is the head of state, the US has a (very strong) Presidential system, where the President is head of government (as well as having a Presidential election system that, even ignoring the role of the President, inherently favors duopoly more than the two-round French system.)
The only elective part is which "faction" of the party the person will be coming from, i.e "Zürich"/"Bern for UDC/SVP.