To clarify first, the intended meaning of the sentence is not :

My partner's score

(meaning the score of my partner), but would be

My score of partners

(meaning the amount of partners, if you prefer).

Would it be correct to say :

My partner score

Or

My partners score

Let's say I made a contest to eat bananas in one day. I ate 12 bananas today. That would be:

My bananas score

Or

My banana score