In character styles / paragraph styles , there are properties like underline, no break, etc that you can set their status to "-" so they inherit from parent style, however, I can't find any method to indicate that I want my character style to inherit color from parent style, or when there's no parent, just not touch the color and leave it as it is.
Thanks for your help in advance,
It seems to be that once you define a color in a character style, you can't change it. The only way to let the 'parent' color fall through is not to define any color in the character style. But if you select text, and then select to create a character style the attributes of the selected text (including its color) will come along into the character style.
So you have to create a character style on a 'clean slate', without having any text selected as you create it and leave the color settings untouched. You can also reset your current character style by making sure that it's 'Based On:' [None] (under General Settings) and then select Reset to Base.
To inherit any attribute of the hosted paragraph style, leave the field empty. If you have defined it, you have to reselect the same color by holding the command key and klick again on the swatch, or click on any swatch first, second with the command key (windows ctrl key.)
Be careful not to overdefine a character style.
If you want to reset to any parent style from the same category you have to click to reset to base. But this deletes any done changes, not only the one single attribute you want.