I know that the monarchy in Great Britain shifted over time from a more absolute form to the constitutional monarchy/parliamentary democracy that now exists. And it's my vague understanding that the greatest part of the shift happened in the last ~200 years.
Where in the "process" was the monarchy of Great Britain at the time of the American Revolution (1765-1783, if we need to be specific)? What level of control did King George have at the time, and how much of it was in the hands of the Parliament?
The short version is 'Control? Officially, not a lot'. The Bill of Rights was established in 1689, which, following on from the Magna Carta and other legislation, set formal & agreed limits on the power of the monarchy both to tax and to rule. The 1689 Bill also laid out the rights of Parliament, free elections, freedom of speech, etc. In 1697, to provide for the Royal Finances, the Civil List was established. Funding was then provided by Parliament to the King, to defray the costs of what was still seen as under the Kings control. So, paying for the Royal household, certain salaries and pensions and more. Now, in 1720, the South Sea Bubble occurred. This was one of the major financial crashes of the last 500 years, ranking alongside with the 2008 crash and the Tulip debacle. The Royal Finances were greatly affected by this disaster, to the point where certain of the King's (George II) financial obligations could not be met. This was a difficult time and required some pretty deft political manouvering on the part of those in power. So, in 1760, the year in which George III was crowned, Parliament put the final stake in the heart of Royal independence and took over pretty much everything to do with the nuts and bolts of running a modern state. However, due to various aspects of parliamentary matters, the position of the Prime Minister was not as well established as it is in modern times. This allowed George III considerable leeway within the political vacuum, and he was able to act in many ways as a First Minister, which is why, from a historical perspective, you see so many references to his role - and his strong opinions - in the 1776 War of Independance. This inspite of the British Parliament creating commissions and sending them off to the US at the time, in the hope of calming things down. After the loss of the colonies, George received considerable criticism and a fair portion of the blame for his involvement and handling of the entire crisis. This reinforced the belief among people of influence that the Royal Family should continue to have less and less to do with the actual running of the state. A view which, sadly, was reinforced in light of the behavior of his successor, George IV, who was not noted for his sobriety or his financial restraint. There's a lot to take on here, and it was a critical time for Royalty and the country. You would do well to research 'the civil list', 'Royal Prerogative', and the various items I've mentioned, if you want to understand more about this time.