The Puritans and Parliament
When Elizabeth first came to the throne, the twelve or so returned exiles in the House of Commons were influential enough to force Elizabeth to make greater concessions to the Calvinists than she may have intended.
But after the initial settlement the Queen forbade Parliament to initiate religious legislation of any sort. (Later she even commanded that religion not be discussed, thus exercising her right to set limits on "freedom" of speech.) Although the Queen's ruling against legislating religion was ignored time and again, bills that passed both Houses were promptly vetoed. These included a 1571 bill to reform the Prayer Book and several later bills to enforce stricter observation of the Sabbath.
As the Puritans became better organized, they had some success influencing elections; but their motions in Parliament to adopt the Calvinist Prayer Book twice failed, and a petition to allow Puritan freedom of conscience was lost in Church bureaucracy*.
Censorship
Despite good organization and brilliant pamphlet campaigns, the Puritans could make little progress against Elizabeth's resistance to change; in 1586 Star Chamber decreed the establishment of the Stationer's Company, which was empowered to censor all writings before they were published and to hunt down unlicensed printing presses.
The Puritans eventually achieved a measure of victory for their cause. Click to read more* about it.
Footnotes
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"Busy fellows"
To her bishops, Elizabeth described the Puritans as "curious and busy fellows (whose) preaching tendeth to popularity." She also directed that such men "be brought to conformity and unity: that they minister the sacraments according to the order of this Realm and preach all one truth."
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James vs. the Puritans
The Puritans shifted their movement underground, only organizing another concerted effort for reform in 1604, with their petition to James I at the Hampton Court Conference. But James defended the episcopal system of church government against a request for presbyters (he is reported to have stated flatly, "no bishop, no king"), and although he agreed to moderate reforms, such as limiting the power of the ecclesiastical courts, they were later foiled by the bishops. Neither Elizabeth nor James achieved a satisfactory compromise with the Puritans.
Charles I's despotism, and his tendency to favour Catholics, increased support for Puritan demands in Parliament, so he determined to rule without it; after 11 years of stretching every legal means to raise money, war with Scotland in 1639 forced him to call the Long Parliament--placing him at its mercy.
Radicals led by the Puritan John Pym drafted the Grand Remonstrance, which granted Parliament control of the army and of government appointments; many moderates in Parliament were driven to side with the King, but the Puritans achieved victory in the civil wars that followed (1642-48).
Picking the wrong fight
The war with Scotland resulted from the King's unwise attempt to enforce an episocopal church government in Presbyterian Scotland.