Oh my goodness gracious..seriously? 4 kids or 2 kids or 10 kids..they have other plans and were clearly involved in something else when you called as you said it sounded like they were at a sporting event for another child. Yes you do seem to want them to alter their plans because clearly you know you called while they were involved in some activity. They did not call you right back thus indicating (to me at least) they are still busy and unable to return the phone call. You want them to call back without seeming to understand that YOU interrupted THEM, you disrupted their family activities and that your phone call is likely very low on their list of things to do for the day. I know it would be very low on my list especially if I had a lot going on that day.
The answer is no..for all the reasons that I mentioned and others mentioned. You called a parent who was clearly occupied at the time at the last minute for a play date that same day. If they have not responded I would take that as a good indication they are busy and thus leave them alone and not call or text them and further interrupt their time/activities.
I wouldn't feel like someone interupted my family time if they called while I was at a sporting event (heck, on the weekends, that's where I always am). If I didn't want to be interupted, I wouldn't answer my phone. I doubt the dad would inform the mom that it was the OP's dd's birthday - I'm sure it left his head a second after he read about it.
Now, as someone with a CRAZY weekend schedule, I can totally see me rushing about, saying I'll call back in 5 minutes, and totally forget. I understand why the OP wanted a callback - she just had to decide if she should call another friend.
This is the reason that, when my kids call their friends for playdates, DO NOT leave a message, so that they can call someone else (same day playdates are frequent here, because everyone lives within a few minutes drive of each other). Of course, people are busy, so you might need to make several calls.
I saw no mention of a party in any of the OP's posts.
Why didn't you take her somewhere fun? Or call someone else?
to your DD.