I don't quite know Bingley, but I am thinking, neither.
I would either use:
A partner's birthday, or
One of the partners' birthdays.

If it happens to be the birthday of one of the partners, then I would use the article 'A' in front.
When I use, 'One of', I am picking one out of the collective many and in this case, the collective is represented by the plural, partners.
An apostrophe, to signify the possessive case, will be out of place in this context, I think.

Shall clock in again at a better time and see if what I just said makes sense.