Actually, the movement that led to the Revolution was not so much Parliament believing that the King had moved too far toward absolutism as it was Parliament itself asserting more and more authority for itself in political matters. Parliament for some years had evolved into a debating society which saw itself as the guardian and protector of the rights of the English people. Sir Francis Bacon, a royal minister, was impeached by Parliament in a display of its position that the king's ministers were answerable to Parliament. Whereas as late as Elizabeth I, Parliament had been denied the right to discuss foreign policy unless invited by the monarch; Parliament assumed this right by opposing the marriage of James I's son Charles to the daughter of Philip IV of Spain, arguing that such marriages were affairs of state, and such a marriage would constitute a pro-Spanish policy which Parliament opposed. Although a showdown seemed imminent, the marriage fell through and the political drama was averted.
Rather than the monarch attempting to gain more power; the opposite was true with Parliament claiming more and more prerogatives to itself. When Charles I asked Parliament for funds, he was forced to accept the Petition of Right which stated that the king could not impose "loans" without Parliament's consent and further that any "gentlemen" who refused payment could be arrested without a show of just cause.
Parliament's increased demand of power for itself and the people led Charles to attempt to rule without it. He was forced to recall Parliament because of a war with Scotland. As a result of the war, Charles was forced to recall Parliament which sat for twenty years, abolished the Court of High Commission through which Archbishop William Laud of Canterbury had enforced Church doctrine, impeached Laud, and passed the Triennial Act which required the King to call Parliament not less than every three years. The situation devolved into the English Civil War as a result of which Charles was arrested, tried for treason, and executed.
Oliver Cromwell ruled briefly as Lord Protector, but the English people were happy to see the Crown restored under Charles II. Charles managed to stay on the good side of Parliament, but his brother and successor, James II, was not so discrete. A devout Catholic, he appointed Catholics to high government positions and dismissed non-Catholic advisors. When his wife became pregnant, he unwisely declared the child would be a boy and would be a Catholic king of England. This was too much for an influential group of Tories and Whigs, who offered the throne to James daughter from a previous marriage, Mary, who was married to William of Orange. They accepted and James was forced to flee. This was the Glorious Revolution of 1688. All these important events were not so much a result of the King claiming more power; but rather resulted from Parliament asserting more and more rights on behalf of the English people.
The civil war and the Glorious Revolution were caused largely by disagreements between Parliament and the monarchy as to the degree of power that the monarch should hold. These disagreements were exacerbated by religious conflict.
When James I came to power in 1603, he wanted to take more power to himself than the Tudor monarchs had. He believed in absolute monarchy under the divine right of kings and he acted accordingly. His son, Charles II, continued with efforts to increase royal power. The two monarchs also wanted to keep the Anglican Church's episcopal, hierarchical structure while Parliament wanted to move towards Puritanism.
Eventually, Parliament and its supporters came to feel that the royal efforts to take more power had gone too far. When Charles II was forced to call Parliament into session to ask for money to suppress rebellion in Scotland, Parliament moved to reduce royal powers. This set off the chain of events that led to the civil wars and eventually to the Glorious Revolution.