This is a thought experiment meant to tease out how flexible our system actually is. It’s meant to highlight that there are a number of practices, including the filibuster, that are done by consent and not actually by law.
The idea is rough, so I probably get a lot of things wrong, potentially including my understanding of what a parliamentary system actually is. What I take it to mean is that the executive is selected by the legislature and can be removed at any time with new elections called.
I believe this could (mostly) be done in America without amending the Constitution. How?
For starters, Congress would agree (if necessary) to endorse an interstate compact that would see all electoral votes voided, effectively nullifying the existing presidential election system (without amendment). It’s unclear whether the National Popular Vote Interstate Compact (currently at 196 electoral votes from 15 states plus Washington, DC) is constitutional without congressional approval, but it almost surely would be with. In the worst case, the states would choose to appoint electors that accede to the parliamentarily-chosen prime minister.
Due to the Presidential Succession Act, the Speaker of the House is second in queue to be president. Thus, the House could become the primary body of electing the equivalent of a Prime Minister. (The PSA is mere law, so if the Senate wants their say, it could be modified to hold the parliamentary-style election however is wanted.)
One obstacle is term limits. Since 1900 at least four people have served more than eight years (cumulative) as prime minister of the UK, and the maximum one can serve as president is ten years (two elevated from vice president, eight as president). While the system could probably milk ten years for any one person as an American Prime Minister, it would require some funky business to install them as VP prior to starting their fifth year, and letting them then succeed a vacancy.
Another obstacle is the regimented lengths of terms for Congress. It would require mass-resignations and states to issue elections as a result. But it is feasible, and in the cases of both the Prime Minister and the Congress, the power to remove anyone who refuses to abide by such a system wouldn’t prove difficult. Issues like legislative seniority are by rule alone, so they could easily be modified to fit a new system.
All of this would be an unusual parliamentary system, in that the head of state and head of government would be the same, unless some new head of state position were created separate from the president. It also wouldn’t transform Congress into a true parliament—the separation of powers are part of the Constitution and would require amendment to change.
But the UK’s parliamentary system isn’t formal—it’s without a written constitution. Perhaps a parliament-by-assent would be in line with that practice.
If the USA ever decided to move toward that style of government, it would amend the Constitution to do so. But there is a lot of flexibility in practices. It could act as a sort of parliamentary system, if it chose to, without touching the Constitution.