Messaging you asking what’s your problem is pretty weird in my opinion. Especially if phrased that way, really off putting. I would give a smile and a hello when you pass them by. Maybe say you take the opportunity before the kids finish to check emails or do some sort of catch up. Maybe even audio book time. A little fib can stop the awkwardness for you and simmer them down. But don’t be surprised if you are a topic of their goss here and there, they sound bored.

I don’t normally socialise with parents at
School pickup , maybe when drop off just a quick hello. You can choose to keep in touch with them by social media casually or just come up with an excuse that your child doesn’t want you to wait for them outside and prefer for you to wait in the car?

In my opinion this is not normal, those parents sound catey and I would be the same and remove myself from that situation. For that mum to message you constantly and ask you what’s wrong is ridiculous and you need to put her in her place and ask her to stop messaging you if it is bothering you. Unless your child is good friends with this mums kid I wouldn’t worry too much as people are pathetic and need to get over themselves. Lots of parents in my daughter’s class don’t interact at pick up and that’s perfectly ok as we all have other stuff going in. I know when I have had a crap day at work, all I want to do is stand in silence and wait for my child to finish