Both mom and dad on the log book but they are now divorced

The log book (V5C registration certificate) does not confirm ownership of the vehicle. It just tells who the registered keeper is. And in most cases, only the name of one person should be listed not now but some people use the first line for firstname to put one full name (eg husband) and the second line for surname to put the second full name (eg wife).

Anyway, coming back to your question, if the finance was in your dad's name only, then he will be able to sell the car. What I mean by this is if your dad has a sales receipt which shows his name printed on, then it is proof he is the owner regardless of what the log book says.

However, with divorce comes the splitting of possession between husband and wife and car is an asset too but that's another discussion...