It is not a major part of life or conversation. If most people think about it at all, there's an acceptance that the resolution was ultimately fudged like everything else in our constitution.

Most people who know a bit about it, if pushed, would probably say they believe Parliament was in the right, but there is at a public level an almost complete lack of understanding of what the issues in question actually were or who actually "won", even among those who have a rough idea of dates, personalities, etc. (For instance, "democracy" was not something either side was actually fighting for or even really thought was a good idea). I think people are also largely unaware that there were three Civil Wars in England from 1642 to 1651 (not counting wars in Ireland and Scotland) and that all of them were important in their own way, but that the sides involved changed and that the Parliamentarians who started the war in 1642 were not the winners in 1651.

In fact the decisive event in English history was not the Civil Wars themselves. Inasmuch as that was Parliament vs Absolute Monarchy, both sides lost, and we got an Absolute Dictatorship. After that fell apart, we then got an absolute monarchy reintroduced, albeit the king in question was very much aware of the real limitations of his power and the need for political consensus, so was careful not to push things too far. His successor did push things too far, was also overthrown and a more constitutional monarchy followed. That's the Glorious Revolution, which is perhaps slightly less well-known than the Civil War overall but the people who do know about it have a better idea of what happened and why.

The Civil War comes up very very occasionally in politics but usually buried in obscure debates and legal cases. It's not something that you hear political figures invoke in their public-facing addresses or rhetoric, nor something people appeal to when campaigning.

The only time it's seriously come up lately was during the Johnson government when there was a tug of war between the government (the "crown") and Parliament over Brexit and Johnson attempted to circumvent the need for Parliamentary approval by getting the late Queen to dissolve Parliament and allow a deadline to expire. The courts ruled that his doing so was illegal, with its being observed that to permit this would be "to reverse the result of the Civil War". As above, I think the learned judge was actually slightly off-base with that, although constitutionally it was the correct decision overall. The outcome reached was to say the Queen had been "badly advised" and so the dissolution was reversed - a fudge that ironically could have come directly from the era of absolute monarchy!

Politics has changed so much since the Civil War(s) that the issues being fought over then are no longer generally considered relevant to modern life. In fact I think the issues of executive vs legislature and of religious tolerance are still very relevant and perhaps the Civil War could do with being thought about more and more deeply than they generally are... but I am both an historian and something of a Civil War specialist, so of course I would think that.