Your clarification makes a much more interesting game.
I think the eight combinations I listed are still the only winning combinations. And I think the winner must have either 3 odd numbers or one odd and two even numbers (and possibly more in either case). So it seems to me that the first player has the best chance of putting together either of these cases. So far, in the test cases I have looked at, it seems that if both players play rationally the game will end in a draw.