Final answer:
King Charles I's dissolution of Parliament in 1629 is the action that drove many Puritans to emigrate to avoid religious oppression, as it signaled his intent to rule without parliamentary approval, alienating and oppressing Puritans.
Step-by-step explanation:
The action taken by King Charles I in 1629 that convinced many Puritans their only hope of avoiding religious oppression in England was to emigrate was when he dissolved Parliament. Charles I, who followed his father to the throne in 1625, was viewed with suspicion by the Puritans, not least because he married a Catholic princess and allowed the Archbishop of Canterbury to introduce more Catholic liturgies into the Church of England. By dissolving Parliament, King Charles I aimed to rule without its interference, a period known as the 'eleven years' tyranny', which included ruling through arbitrary and non-parliamentary measures like imposing taxes.
This action alienated Puritans both within and outside of Parliament, leading many to seek a new life in the Americas, establishing colonies like New England as a haven for reformed Protestantism.