Maybe this has to do with how frequently we come across SAHDs...? To me, this is just a no-brainer that you treat the dad just the way you would treat a mom. At least to me, it would feel like a discredit to me, my marriage and my partner if another woman didn't hang out with my partner because of a presumed jealousy problem. For me the whole mom/female solidarity thing (which I value) would be expressed by assuming that this woman speaks with her partner and everyone is on the same page.
And, if you have a pretty diverse group of friends, this kind of thing will continue to become an issue, no? It has for me. My oldest is 12 and in most of her friend's families all the parents work out of the home in some capacity. Everyone divides responsibilities differently and when an event is planned and parents are involved it's not like we get to pick which parent attends and in what capacity. I've been on two overnights where it was the father attending and many, many get togethers where the father drove, stayed for drinks and we had a great time. This always makes me feel closer to the entire family.
This is not to say, Lima, that it didn't take a slight adjustment for me when this first happened to us! I still have to check myself that I'm not defaulting to the mother all the time when it comes to arranging stuff. My instinct is to go for the mom's email and I know that's a source of frustration for some families in my community. But, if we all help get out of that mindset -- I think it's better for everyone.