It really depended on where you were. The American Revolution never really shook up the domestic power structure of the American Colonies. The guys who were rich and powerful in North America before 1775 largely remained so after 1781, especially if you consider how radical the social and economic system was altered during the French Revolution. Those at the top of the system favored some reforms and democratizations which were left of what would have then been a Conservative British approach. When compared to continental policies, even the most Conservative members of the Continental Congress earned the "radical" moniker.

But not every radical is made equal, and there was a lot of debate as to exactly what form democracy would take in this new system. Theres a huge movement here in America which suggests that the "founding fathers" got the Constitution right the first time, and that it went from God's mouth to there ears. That really leaves out 13 years of pre-Constitution political development, as well as the post-1789 insurgency against the Constitution.

Here in my state, the great state of Pennsylvania, we hold a privileged place as one of the more liberal and radical states. Prior to 1789, we had a really funny constitution(for clarity, when I say Constitution, with a big C, that refers to the Constitution of 1789. All others are spelled constitution), which had a whole bunch of rights not found in the Federal Constitution. It actually has a pretty good wikipedia page, with a copy of the full text of the constitution. Notably, it included full franchise for all white males who paid taxes (the Federal Constitution started out with property requirements). It also featured a Council of Censors, whose only responsibility was to evaluate the laws passed. They acted much as the Supreme Court does today, in that they decided if a law was oppressive or just. Further, the assembly had no power to pass laws during its term. Instead, any law passed during one session would be passed out across the state, considered by its population, and then (if accepted and not repealed) it would come into effect in the next session (the next year). The PA constitution was all about empowering the common people, getting them into politics, and making sure government's powers were checked. And not checked by another branch of government, but by the direct sovereignty of the people.

The constitution could also be interpreted in some other really cool ways, it made provisions for atheists:

And that no man ought or of right can be compelled to attend any religious worship, or erect or support any place of worship, or maintain any ministry, contrary to, or against, his own free will and consent

And there were even intonations of what we would today call socialist and communist thought, though it was then likely harkening back to the English Civil War of 1649:

That government is, or ought to be, instituted for the common benefit, protection and security of the people, nation or community; and not for the particular emolument or advantage of any single man, family, or soft of men, who are a part only of that community

These truly radical ideas were never embraced by the Philadelphians who lived in the hub of Pennsylvania. But out on the frontier, people were far more interested in the notion of Pennsylvania as a Commonwealth.

This constitution was pretty popular with the lower sorts, and even some of the more established elites across the state. But after 1789, the states all moved to align their constitutions with the new Federal Constitution. Here in PA, the Constitution of 1790 altered the political balance. Many of our rights were altered, especially considering the Censors, who were removed. Their powers wouldnt resurface again until 1806. This made sense, federal policy and state policy entered sync, but the Federal Constitution represented a major step back in some areas, again like with the Censors and with the franchise.

This really clashed with PA's longstanding liberal attitudes. Many had fought in the Revolution, and now they saw the federal government stripping them of their rights and changing the political structure against the popular will. And one of the biggest ideals in the old '76 constitution was the idea that only the people had the right to change, alter, amend, or abolish the current political order. For those who didnt agree with Federal policy, for whatever reason, the new constitution seemed to violate that principle.

This actually resulted in several rebellions, including the Whiskey Rebellion, and the lesser known Fries "Rebellion" (which Ive written a paper on, and done some research on). But to really quickly sum up both rebellions, they were really just a bunch of pissed off farmers, who wanted to be left alone, and were mad that the Federal government was instituting a tax that they didnt vote for. In the case of the Fries Rebellion, the constitutional challenge was even framed in the language of the '76 constitution, with many of the "rebels" arguing that the Federal government had no authority to enforce laws less than 1 year old! Basically, their more liberal ideas clashed with the Federal Government, which itself was trying to end the Revolution.

I cant really speak to how this fits into the British Parliamentarianism of the period, as Ive never really studied that, but its important to remember that the political order of the colonies was in flux between 1775 and 1789. In some places, the revolution became somewhat experimental, while in others it remained very traditional. The Constitution of 1789 was so hard to write, and required so many of those famous compromises, because everybody had developed a way of doing things, and they thought their way was the best. The Constitution was never going to please everybody on every issue, but it was absolutely necessary to end the experimentation of the Revolution. Its ability to bridge the gap between the conservative and liberal elements of the country led to its tremendous fame and success, and it succeeded in creating stability without much bloodshed. But it was always a middle road between the left and the right. Unlike the French Revolution, when liberals and then monarchists took complete power, the American system was forced to "thread the needle" between extremely different political perspectives.

Edit: The 1776 PA Constitution isnt actually on that Wiki page, just the Declaration of Rights. The rest can be found on the Avalon Project

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Sources:

Terry Bouton, Taming Democracy: "The People", The Founders, and the Troubled End of the American Revolution

Paul Douglass Newman, Fries's Rebellion: The Enduring Struggle for the American Revolution