This might sound harsh but if your sister stole $10G, she stole $10G. Actions have consequences, and consequences can be a blessing if they help a person see that doing XYZ to get through life "doesn't work." Sparing her consequences is teaching her that she can get away with stuff like this, especially if she plays the victim.
Your mom had nothing to do with it and has no obligation to pay your dad, so it's up to your dad whether he wants to cover that charge or have the credit card company investigate and let the law decide the consequences.