The odometer is digital and does not rack up miles in the Accessory position. I tow a 2007 CR-V (same as your 2008) and it does not rack up miles. It only counts miles in the Run position. I towed a 1999 CR-V for years and many of my friends two CR-Vs of various years and none of them count miles while towing.
If your does rack up miles in the Accessory position your CR-V has a problem and needs to be returned to the dealer for repairs.
2005 Dutch Star 4015
2007 Honda CR-V