Do some British people still refer to their parents as "Mater" and "Pater"?

Only by way of a joke.  A few very upper-class people may have done so 50 or more years ago, but certainly not since then.

Of course not, except as a joke. It wasn't common years ago either, even among the upper class.

There may still be a handful of such people. But it's certainly not at all common any  more - not for the last 100 years.

Actually I have heard this in period movies and read it in British movies.  In aristocratic families only.  It seems to have been done still in the 1950s, but even that is a long time ago.  So I was just wondering.

I live in the UL and am British. No, I know of no one who addresses their parents in Latin nor have I heard of any doing it. I have no idea how you got the idea we do this. Haec inepta quaestio est.

I never did. That would have been known as 'being affected' down our way.

Maybe in some unworldly upper-class families in the boondocks, or used humorously.

Only in paternity and maternity.

I’m British and have never heard of that 

Nobody has ever said that seriously!  I say pater quite often.  It's always tongue in cheek.  Yes, I did go to a boarding grammar school (that's like a prep school in USian).