Mary,

uni=one (eg unicycle as opposed to a bicycle);
uni-rig must mean one sail

So you're right in what you're saying. But in our realm "unirig" is often used synomymously for "catrig." An A-cat is really a unicatrig. When a monohull drifts along under jib alone, most of us don't think of it as a "unirig" even though this is linguistically correct. But any boat sailing under main alone (ie one sail positioned aft of the mast) is sailing cat rigged.

So Parlier's boat has a twin catrig...or maybe a bicatrig rather than a unicatrig.