(Edited)
You want scientific basis on the effects of not having a rest day? Here's citations from the Wikipedia article on overtraining:
http://en.wikipedia.org/wiki/Overtraining. Since most of these are from medical journals I think they should suffice in providing scientific justification for having a rest day.
^ Roose, J; De Vries, WR; Schmikli, SL; Backx, FJ; Van Doornen, LJ (2009). "Evaluation and opportunities in overtraining approaches". Research quarterly for exercise and sport 80 (4): 756–64. PMID 20025117. edit
^ Jeremy Adams, Robert J. Kirkby (1998), "Exercise dependence: A review of its manifestation, theory and measurement", Research in Sports Medicine 8 (3): 265–276, doi:10.1080/15438629809512532
^ Draeger J, Yates A, Crowell D. 2005. The obligatory exerciser: assessing an overcommitment to exercise. Pys Sportsmed. 33(6): 13-23.
^ Dave R. Baldwin (2002-03-27), Exercise Motivational Triggers, iUniverse, p. 53, ISBN 9780595216031
^ [Bresciani G, Cuevas MJ, Molinero O, Almar M, Suay F, Salvador A, de Paz JA, Marquez S, Gonzalez-Gallego J. 2011. Signs of overload after an intensified training.International Journal of Sports Medicine. 32(5): 338-343.
http://www.ncbi.nlm.nih.gov/pubmed/21380974 ]
^ Sharp, NC; Koutedakis, Y (1992). "Sport and the overtraining syndrome: immunological aspects". British medical bulletin 48 (3): 518–33. PMID 1450881. edit
^ Bresciani G, Cuevas MJ, Molinero O, Almar M, Suay F, Salvador A, de Paz JA, Marquez S, Gonzalez-Gallego J. 2011. Signs of overload after an intensified training.International Journal of Sports Medicine. 32(5): 338-343.
http://www.ncbi.nlm.nih.gov/pubmed/21380974^ Smith DJ (2003), "A framework for understanding the training process leading to elite performance", Sports medicine 33 (15): 1103–26, doi:10.2165/00007256-200333150-00003, PMID 14719980
You want non-technical citations for people who aren't physicians? The about.com article are pretty good.
Training rest DayOvertrainingAs to what you should do? Well, I'll be interested in seeing what the skate moms have to say.
If the hours are to the extent that the child feels abused then that's an interesting question. Mention it to the skating director and let her investigate and handle it behind the scenes? It depends on the director and the coach I suppose. I thought about this and decided I want to withdraw the idea. There's a lot wrong with it.