I know it is confusing for Americans who are not familiar with Parliamentary systems. In my very humble opinion, your Republic is a superior system in the aim to reduce the power of government and secure the freedom of individuals - however, the Parliamentary form is far superior than the models that most European nations are built on. One exception may be the Swiss Confederacy. The Cantons have tremendous amount of power and independence. The one problem I have with the Swiss is the requirement to serve in the military. That seems to be a violation of individual rights.
Most European nations do have a parliamentary form of government. Though republics w/ a parl. sys., the two notable exceptions would be France and Russia. They're semi-presidential; i.e. parliamentary-presidential but their respective presidents have considerable executive powers. Russian president even has the power to dissolve the Parliament at his own discretion.
Presidents in other Euro countries have a largely ceremonial post (i.e. representing the state internationally, receiving visiting dignitaries, cutting the ribbon, etc.).