Yup - I've read the thread. Obviously people would make adjustments for the child's special needs, but the OP started this thread about parents who want to hang out at her house, not a particular situation where one child has an additional need, or overwhelming shyness/lack of confidence. She was talking about a mother wanting to come for a chat for 2 - 3 hours.
Also - these children are in Yr1 - they are not Pre-schoolers. They've been going places without their parents for a year and a half now, at least, most probably at least a year more than that.
I could ask, have you read comments from StandupMop, Seeline, Minty, GlitterMud, redskybynight, etc? For everyone I know in RL (as I said above, I have actually been round and asked people after being amazed at threads like this before), having a child round to play after school, means they play with each other, and each parent gets on with their own life, in their own homes.
Oh! I've just remembered one time when a parent did come - her family had just moved to this country, and her ds had no English yet, so she asked if I'd mind if she came with him. Of course it wasn't a problem, as the reason then was that it enabled the ds to come. But that was an exceptional circumstance, like those you mention, not just a parent being a pain.
Obviously, if you want to meet with a friend of yours and the dc can play together at the same time, as some posters have said thy do, that's lovely, but it's not the situation the OP is talking about.