Is parliament a part of the legislative branch?

This is a more complex question than it might first appear. Parliaments do indeed legislate, but unlike the American system of government, in which the President is elected separately from individual Senators and Representatives, most parliamentary democracies elect their executive (Prime Minister) by allowing voters across the country to choose their local MPs (Members of Parliament). Thus the separation of powers (and branches of government) is not as distinct in a true parliamentary democracy. 

That is because voters choose their MPs based on party affiliation; they vote for whichever political party supports the policies they want enacted. In England, for instance, if a voter from the industrial city of Manchester supports the Labour Party, he or she will vote for his or her local Labour MP. Yet the Labour party will have already selected its leader, so a vote for a local Labour Party MP will serve as a proxy vote for the national party, and thus the Prime Minister (executive) is not directly elected in a popular vote. When a political party wins a majority or plurality of seats in the parliament, that party's leader becomes Prime Minister.

So while the parliament does indeed legislate, it also serves the functions of the executive. For instance, the prime minister selects the members of his or her cabinet directly from the parliament, and the MPs who comprise the cabinet do not leave the parliament, but remain part of it. They simply become part of the governing coalition, and therefore have more sway than other MPs who are not in the cabinet.

Essentially, the parliamentary system of government has a series of checks and balances built into it, but the "branches" of government are not distinct. Therefore, there is no proper "legislative branch," but there is a legislative body.