The most prominent (and best looking) examples I can think of in modern games are Arc System Works' more recent fighting games, the
BlazBlue series and
Persona 4 the Ultimate. Both games use all 2D (pixel) art for all the characters and most effects (hitsparks, misc particle effects, etc), but the backgrounds are completely done in 3D. They use a nice mix of polygonal geometry and 2D assets (mostly for characters in the background), but the overall look gels really nicely with the characters and makes for much better depth in the backgrounds than you'd tend to get with pure 2D backgrounds, even with lots of parallax scrolling.
(Other 2D fighters that used the same technique to varying degrees of success include Capcom vs SNK 2, Marvel vs Capcom 2, King of Fighters 94 Re-Bout and The Rumble Fish)
Another example that quickly comes to mind is
Strider Hiryu 2. It doesn't look quite as impressive on accounts of running on more primitive hardware, of course, but it still looks nice and I think the use of 3D serves the game well.
Generally speaking, I think 3D can be used to great effect in 2D games, if done tastefully and with some thought behind it. Basically as long as the camera remains at a fairly static angle relative to the sprites I think 3D can add a sense of depth to something you would expect to look kind of flat, whereas 2D elements in a fully 3D environment kind of look flat where you expect three-dimensionality. Regardless of how you do it though, I think a key factor is to make sure that the 2D and 3D elements simply fit together as well as possible so that neither looks out of place compared to the other. I think the Pokemon and Prinny games mentioned earlier pull this off quite well, judging by those screens, whereas... I dunno, Mario Kart 64 perhaps doesn't manage to do it quite as well. Strider Hiryu 2 arguably has a more visible discrepancy between the 2D and 3D elements, but I dunno, personally I think it works quite well anyhow.
I guess it's a matter of taste; I like 2D art, particularly well made pixel art (and it doesn't get much better than Capcom in the late 90's), so in some cases it doesn't bother me if the 2D elements call a certain amount of attention to themselves.