These claims have all been made in the UK Larry (yes we have our share of home grown nutcases at both ends of the political spectrum,), and BTW the right side, i.e. The Parliamentarians, won the civil war. King Charles 1 had let power go to his head and so was relieved of his head.
Life under puritanism was hell of course, they even banned Christmas — and that is a documented fact — along with football, dancing and theatre. Fortunately the Royalists, duly chastened by their years our of power during the interregnum, managed to effect a restoration that created a kind of power balance between parliament and the crown.
The problem with “the other side” as you call them lies in the fact that they assume too easily the rest of the world is as obsessed with America’s partisan and parochial arguments as they are.