I don't believe there's any way to structure that sentence with "John Smith" and "my father" in that order to express possession of the car. It would have to be reworded so that the noun phrase and apposition were the possessed object, rather than the possessor with "'s" alone:
John Smith's car -- my father's car -- is red.
As to your two examples, the first one is out because you can never say, "my father car". The second one is closer, but doesn't work either because the "my father's" is an adjective phrase, so it cannot be in apposition to "John Smith", which is a noun phrase.
The closest I can come to acceptable is to put possession on both nouns:
*John Smith's -- my father's --(??) car is red.
This doesn't work either in writing or in casual speech. In writing, you need a dash (--) before "my father's", but you cannot have a dash after it (hence the ?? above) because it would visually break up the noun phrase "my father's car". You also can't have just one and not the other.
In speech, there are no punctuation marks, but if you put the possessive on both nouns, it sounds like you're correcting yourself or revising your sentence to begin with "My father's...". A listener would not likely understand the second to be an appositive.