In OpenTTD, it's possible to build multi-platform train stations side-by-side (pictured below as Parnpool Mines), and they would work as expected, allowing two trains to load/unload at that station at the same time.
However, sometimes due to space constraints it would be more desirable to to build the two platforms lengthwise as shown below in Parnpool West. The signal-station combination pictured below, however, does not work, and I have not found one that does.
Best Answer
There isn't any solution that I can see after puzzling over the image for a while. The only ones that would work would be splitting them by two tiles instead of just one and using a bypass track to get around it, but that would mean you're going up to 2 tiles of width anyway, rendering the issue moot.
The only workable idea that I can come up with is to set them as two different stations and use the "Non-stop" flag for the two different trains (train A to get it to ignore the first station, train B to get it to ignore the second station) but then if your trains happen to get out-of-order you're pretty much up crap creek.
I guess the only real answer is that there is no answer.