Disney's Aladdin JR.
Disney's Aladdin KIDS
Annie JR.
Annie KIDS
Disney's The Aristocats KIDS
Disney's Beauty and the Beast JR.
Disney's Cinderella KIDS
Dear Edwina JR.
Magic Tree House: Dinosaurs before Dark KIDS
Elf The Musical JR.
Fame JR.
The Musical Adventures of Flat Stanley JR.
Disney's Frozen JR.
Magic Tree House: A Ghost Tale for Mr. Dickens JR.
Guys and Dolls JR.
Disney's High School Musical 2 JR.
Disney's High School Musical JR.
Honk! JR.
Roald Dahl's James and the Giant Peach JR.
Disney's The Jungle Book KIDS
Magic Tree House: The Knight at Dawn KIDS
Disney's The Lion King JR.
Disney's The Lion King KIDS
Disney's The Little Mermaid JR.
Madagascar - A Musical Adventure JR.
Mary Poppins JR.
Disney's My Son Pinocchio JR.
Magic Tree House: Pirates Past Noon KIDS
Seussical JR.
Seussical KIDS
Shrek The Musical JR.
Singin' in the Rain JR.
Disney's Winnie the Pooh KIDS

Main restrictions: video recordings are for cast/crew/families only, can't be sold and can't be broadcast in any way including on the Internet.

The remaining shows do not allow videotaping. Parents cannot tape, even just for their own personal in-home use. It has nothing to do with how the tape would be used, not being sold, etc. It all comes down to legal copyright rights. Here is how Music Theatre International explains it:

Unfortunately, the rights to videotape are not available. Copyright law gives authors the exclusive right to control the reproduction of their work. When a license is granted for a live stage production of a show, that license does not include the right to tape it because the authors retain the sole right to decide when or if their work is recorded in any way. Even a videotape made for classroom use, as a personal meme