Parliamentary democracy fuses executive and legislative power rather than separating them: the head of government, usually called a prime minister, is drawn from the elected legislature and stays in office only as long as that legislature is willing to support them. Lose a vote of no confidence, and the government falls, often taking the legislature itself down with it into a fresh election.
Britain built the model gradually rather than by design. The constitutional monarchy that followed the Glorious Revolution left day to day governance to ministers, and over the eighteenth and nineteenth centuries the convention hardened that those ministers needed the confidence of the House of Commons to govern at all. The Bill of Rights of 1689 supplied the legal foundation without ever mentioning a prime minister or a cabinet by name; those offices grew out of practice, not statute.
The model spread widely and adapted freely. India's 1950 constitution paired parliamentary government with a president who holds the head of state role split from the head of government, elected not directly by voters but indirectly, by an electoral college of the elected members of both houses of Parliament and of the state legislative assemblies. Japan bound its postwar parliamentary system to a constitutional monarchy. Whatever the local variation, the defining feature holds everywhere it appears: a government that governs only for as long as, and only because, the elected legislature lets it.