Typically the answer is "No." If your sponsor is in a reserve or guard unit making drills and annual training then their dependents can not fly Space A. The sponsor can fly within the limits as per Table 3 of the DoD regulation. If your sponsor is a retired reserve (guard) not age 60 then dependents cannot fly Space A. The sponsor can fly as per the regulations (see "Gray Area Retiree FAQ below). If the person is over 60 and holds a retiree ID card (blue) and the dependent holds the DD Form 1173 dependent card then the dependent can fly Space A when accompanied by the sponsor. They are treated exactly like regular retired and can fly Space A anywhere there are flights going.
If your sponsor becomes activated for more than 30 days then the typical answer of "No" changes to "Yes" and the dependent has all the Space-A privileges of an Active Duty Dependent accompanied by their sponsor on ordinary leave (except dependents cannot travel unaccompanied under the Command or non-command sponsored travel programs). IMPORTANT! - Make sure all dependents of an activated Guard/Reserve Member update their ID Cards to a regular DD Form 1173 (United States Uniformed Services Identification and Privilege Card) versus the DD Form 1173-1 issued to Guard and Reserve dependents!