Probably the best chance for the de facto establishment of parliamentary government in the US was pre-1900--the Andrew Johnson impeachment. Indeed, that was seen to be the real issue in the impeachment fight by some observers at the time.
Consider what was written by a friend of J.A. Garfield (quoted in W.R. Brock, *An American Crisis: Congress and Reconstruction 1865-1867*, Harper Torchbooks edition, p. 260): "The next great question to be decided in our history is this--is the National Legislature to be as omnipotent in American politics as the English is in English politics?...May we not anticipate a time when the President will no more think of vetoing a bill passed by Congress than the British Crown thinks of doing the same thing?"
Also note the remarks of Wisconsin Senator Timothy Howe on the Tenure of Office Act: when a Democratic Senator referred to the President's "own cabinet" Howe specifically denied that it was such. It was, he said, "the Cabinet of the people." He compared the American and British systems and said of cabinet members that "it is no more necessary that they should be on confidential terms with the president than that they should be on confidential terms with the representatives of the people."(Brock, p. 259)
I doubt that such views would prevail even if Johnson were convicted. Once Grant would be elected president in 1868 (after a few months interim rule by Wade) it seems unlikely that Congressional Republicans would adhere to such an unorthodox position on legislative-executive relations. To the extent that they came to such a position even temporarily in OTL, it was only due to extreme frustration with Johnson's systematic sabotage of Congress's Reconstruction policy (originally, John Sherman did not even want to include Cabinet officers in the Tenure of Office Act). With a popular president of their own party, they would probably revert to more traditional practice.