In countries with a parliamentary system of government, does the President or Prime Minister have real power?

Your question is deceptive. There are many types of parliamentary models, not one standard. Some have a president, some have a prime minister, some have both, some have neither! It's quite confusing. However, as a general rule, the prime minister is head of government (he runs the internal stuff) and the president is head of state (he talks on behalf of the country to other countries). For example, in the US, the president is both head of state and government. In England, the Prime Minister is head of government and the Queen is head of state. That position is mostly honorary, though...