It’s about how much power the head of state has. If they directly run the government and make policy, presidential (or Prussian Constitutional, in the case of a monarchy). If they don’t and the power is defacto devolved to a parliamentary cabinet appointed by the representatives, parliamentary. If you’re the French and De Gaulle fucked up your government to maintain power, semi-presidential.

The advantage of parliamentary systems is they fail quickly and cleanly. If the governing representatives lose the ability to vote in a budget, they’re not in power anymore (either a new coalition forms organically, or elections are called), while a US system can fuck around for decades with an inability to pass policy. The downside is the difficulty in making and maintaining stable majorities, see Belgium.