Thanks, everyone, for the interesting discussion. In my OP I was intentionally vague about which stars I was viewing. I was more interested in a general method for calculating the distance between stars as viewed from earth. Maybe the question should be: Are stars farther apart than they appear?
Precisely.
Spect Brain*
Are stars farther apart than they appear?
*
Are you using your rear-view mirror as a telescope? 
Anyway, my calculator should solve those angular size and distance calculations.
www.1728.com/angsize.htm
The closest stars to the Sun are the three stars of Alpha Centauri, at about 4.3 lightyears. The closest of these three is a little red dwarf sometimes called Proxima Centauri, which might not even actually be bound to the other two, while the other two are a star similar to the Sun, and one a bit cooler and smaller.
Next after the stars of Alpha Centauri comes Barnard’s Star, at just under six Ly, Wolf 359 (yes, Star Trek fans, that’s a real star) at 7.6 Ly, and BD + 36º 2147, at 8.1 Ly. All three of these are red dwarfs. Then we have Sirius, rather larger and brighter than the Sun, and its white dwarf companion, at 8.6 Ly, then three more red dwarfs, then epsilon Eridani at 10.7 Ly.
Incidentally, all of these red dwarfs are so inherently dim that none of them are visible to the naked eye, even though they’re so close. Any reddish star you see in the sky is a red giant (a completely different sort of star).
The most accurate figures available for stellar parallaxes are probably those listed on the RECONS ONE HUNDRED NEAREST STAR SYSTEMS list[sup]1,2[/sup]. Of those, Sigma Draconis[sup]3[/sup] seems to be the most accurate at 0.17359 +/-0.00041. Most of the parallaxes on that list are accurate to 5% or better. They update their list about once a year.
[sup]1[/sup]Drop the TOP100.html part of that url to get to the RECONS home page.
[sup]2[/sup] Actually the 107 closest systems. In the last update, they left the displaced systems on the page.
[sup]3[/sup]That’s the 66th nearest system. You’d think closer stars would have more accurate parallaxes, but if so, it’s only a very general rule. There doesn’t seem to be much of a pattern in the error bars on that page.