I'm not clear what it is that is happening. You daughter is in 4th grade, right? So, 9 or maybe 10 years old. I'm not sure why your marital status comes up at all, really. I get that a random parent might ask about "your husband", but after how many years at this school, and how many events, don't you start seeing the same parents? Aren't they already aware of your status? So why is it coming up?
And as others mentioned, there is likely some other reason that you are not perceiving. Do you talk too much? Are you trying too hard? Are you defensive when your spouse (or non-spouse) is mentioned? And as another poster mentioned, "separated" is sort of a loaded word. It does imply drama. It implies upheaval. And instability. And private matters. And nobody wants to ask about that.... and probably don't want to appear rude or insensitive, either. It's easier to ignore completely. But if they instigate contact with you, they can't ignore it. Or at least, they imagine it will come up. Will planning a play date with you mean involving this mystery dad who nobody has met?
Sure it's judgy, but nobody has time for additional drama, and nobody wants to involve their kid in in someone else's drama, either.
So. Take a long hard look at what you present to these other parents. It is NOT a package deal. It's your daughter. You are just a chauffer. Stop looking for friends to hang out with among the other parents (sadly, that doesn't look like it's going to happen for you) and make friendships elsewhere unrelated to your daughter's friendships. Sure, it makes a nice tidy package if they could all be the same people. But that is not happening here. So, move on from that idea.
Make your interactions with fellow parents about the children only. Because those are who have relationships with one another--the kids. Save your personal relationship building for your own friendships. Elsewhere.
Maybe in time things will change. But they might not, and you should be prepared for that. Let your daughter invite her friends to your home or out for a movie or playdates or whatever. Let her give the kid a written invitation for whatever it is with your phone number and a note to call to discuss any details or whatever. But keep it focused on the kids.
And as for the dad who no longer picks up his kid, his wife does it now. There is nothing wrong with that and doesn't imply any issues with either of them. They are a smart couple in my book. They are protecting their marriage, by avoiding even the risk of the appearance of impropriety--the start of all rumors. They are avoiding drama.
Which is another checkmark for me that you are projecting something you don't realize you are projecting.