A friend of your dad
A friend of your dad's
are valid (at least in British English).
I think this only works when we are talking about one or more particular members of some group. For example, Pete may have many friends, so we can say
I met Mary, who is a friend of Pete's.
to mean the same as "I met Mary, one of Pete's friends". But assuming Pete has only one mother, we could say
I met Mary, who is the mother of Pete.
but we could not say
I met Mary, who is the mother of Pete's.
It might help to think of "a friend of Pete's" as an abbreviation for "a friend of Pete's friends", that is "someone who is one of Pete's friends".
EDIT: CopperKettle's 'picture' examples also provide a good case for the use of this construction!