My parents will be visiting from England and I want to take them to a restaurant that has real waiters, the kind who quietly take your orders, only appear when they see that you need something, don't sit down at the table with you (my mum would have a fit), don't keep asking if everything is all right and don't try to hustle you out by bringing the bill when you are still drinking your coffee.
Does such an establishment exist?

There was a place called Jacob Wirth's downtown that claimed it was serving meals "before your grandfather was born". It was very old, and so were the waiters. They were all dressed in tuxedoes and they were real waiters as you described, just very slow.

Jake Wirth's is still there - on Stuart St. in the theater district. I haven't been in a while, but the last time I was in I think the waitstaff was pretty much like everywhere else. I do remember the older professional waiters there from many years ago, however.

Any of the more upscale restaurants will have very professional wait staff. Try Grill 23, Hamersley's Bistro, Radius, No. 9 Park, etc.

Unfortunately, many restaurants feel the need to compete with the TGIF's of the world.

Egg......

I'd vote for either the Cafe at the Ritz or Aujourd'hui at the Four Seasons.

Jake Wirth's is indeed still there. I think they have changed hands (or all the old waiters have expired) because the wait staff was younger last time I was there (in March '02). Also, the menu has been upgraded and the prices have gone up considerably. The good news was that my favoriet dish, although more $$ and served in a fancier version, was even tastier than in its previous version. But it's no longer a "good, cheap place" for dinner. Oh well, if it keeps it in business, then I guess it's worth it. One bonus is that they validate parking if you eat there.