Sorry, I don't know the specifics of the parking brake on your Odyssey. You can try jacking up one rear wheel and see if it is free to turn when the parking brake is set.
Yes, the Honda dealers are pretty non-committal on the tow capabilities of their models -- most likely an inoculation against warranty claims, etc. My Acura specifically has a section in the manual on being towed behind a motor home with a detailed procedure to set it up in regard to cycling through the gears to circulate the transmission fluid prior to towing. If a brain dead owner did not do that conscientiously, it might result in more service claims than the manufacture thought they were liable for and so they just quit publishing that the MDX was tow capable after the 2005 model year...
2007 Tour Master T40C
Acura MDX toad
"It takes a great deal of time to recover from any improvement..."