Parliamentary democracies still have distinct executive branches.

Sure, but they are usually appointed by the majority party of parliament and its leader becomes head of government.

No, that's how a dictatorship works, not a parliamentary democracy. The prime minister/head of state may set policy, but it's still subject to an up/down vote in the parliament. In all cases I can think of, a prime minister can't unilaterally make such drastic changes without (parliamentary) referendum.

The U.S.'s federated system just makes it more explicit what's going on.