Well prime-ministers of parliamentary republics are practically equivalent to presidents in presidential republics. There are only three (basic) differences:
*In presidential republics people elect the president, who is head of government and head of state. They isn't member of parliament.
*In parliamentary republics people elect the prime minister who is head of government but not head of state. They is member of the parliament.
Generally speaking "prime-minister" is the head of government in states where head of government and head of state are different persons.
Generally speaking, "president" is an elected head of state.