What does parliamentary army mean?

During the English Civil War, which went on for twenty years or so and ended in about 1648, the fighting was between the forces of the King, Charles I, and the forces of Parliament. The Parliamentary Army was eventually led by Oliver Cromwell, and won the war, after which they beheaded Charles I and Cromwell ruled England for about a dozen years as the Lord Protector, turning down an offer to make him King. Sometimes the army and supporters of Charles I are called Cavaliers (the source of the nickname for the athletic teams at the University of Virginia), and sometimes the Parliamentary supporters and forces are called Roundheads (from the hats they liked to wear), or Puritans. These were the same Puritans who at the same time were being driven from England for disagreeing with the Established Church, which was the Church of England, of which the King (or Queen) is head. These were the settlers of Massachusetts and New England in the northeastern US.