I understand that you are friendly with the mom, but the father has 50/50 custody and he is just as much a parent as she is. So don't leave the dad out, and don't insult him by communicating with the mom when the dad is the one arranging the play date. As you say, the dad has as much of a right & responsibility. We do a disservice to fathers everywhere when we assume they can't, or won't, take charge of child care. Maybe the mom ran the show when they were married, but maybe he's a hands-on dad -- in fact, in some cases, maybe the mom's controlling things is a factor in the divorce! You never know!
In this case, the parents are working well with each other. But even if they weren't, you have to do the right thing, which is to communicate with the parent who dropped off and is picking up the child. If it were to come up in conversation like you suggested ("Hmm, why didn't I know that my son was at my friend's house?"), the answer is, "Gee, you ex Bob dropped him off and made the arrangements during his time, so I touched base with Bob."
When my husband's kids were with us, we often had their friends over, and it was a little more of a hike because we lived 30 minutes away. We always contacted the visiting child's parents ahead of time (and I introduced myself if I didn't know them), did the invitation, told them we would pick up and drop off, outlined the activities (such as sleepover & breakfast & then playing in the sprinkler) so they'd know what to expect and pack, asked about anything like medication or strong food preferences, gave our address and phone number, etc. My husband was always a very involved father (more so than the mother actually) and so it was very hurtful when time with us was limited. It was so important to him that his kids have normal activities while with us, including their friends. I can't imagine what would have happened or what message would have been communicated if the friends' parents had contacted my husband's ex instead of him!