I am curious about Ming Dynasty Chinese swords that require two hands and that have parallel edges; that is to say, swords that do not have curved blade. Are these swords still termed "jian"? Also, does anyone have a photo of an extant antique sword of this sort?

If you mean double-edged swords (there were straight dao, and jian usually have profile taper and therefore don't have parallel edges), then they're still jian. For a more specific name, you call them shuangshou jian 雙手剑 (or 双手剑), literally "two-handed sword".

I don't know of any Ming examples. I only know of one post-Ming example that "requires" two hands. Probably Qing.
http://www.swordsantiqueweapons.com/s1380_full.html
Longsword-sized. Handled this. I'd estimate it to be about 1.3kg, balanced a little further out than a European longsword, but the pivot point is close enough to the tip for good longsword-like handling.

More common are smaller jian with two-handed grips. These don't require two-hands, so might not fit your criteria. Some Qing examples:
http://www.mandarinmansion.com/items/shuangsh...sword-sold
http://www.vikingsword.com/vb/showthread.php?t=5910

It's easier to find pre-Ming examples.

You might find one or two here.

http://forum.grtc.org/viewtopic.php?t=893