So I talked to a very cordial woman in AEA's Los Angeles office and a very helpful man in the New York office. Under the Staged Reading Code, the most basic of staged reading agreements for Equity theaters, parents can basically go fuck themselves—and pretty much anybody who wants to videotape a performance under that code. Exceptions, from what the AEA reps know, are never made.
There is what's called a "Staged Reading Contract" that, under certain circumstances and through some negotiation, allows for the theater to videotape performances for archival purposes—but they aren't allowed to disseminate copies, e.g., to parents whose children wrote plays being performed. Parents or any audience members are never allowed themselves to in any way, shape, or form record the performance. As far as I have gathered, there are no waivers that exist that would allow people to record shows, even if they agree that it's for private use only and that it would never be broadcast anywhere or sold.
Actors, you know I love you, but your union needs a good beating.
—Reporting From Glendale, California