This is an interesting question. I have had similar thoughts regarding this topic with many variations of the same situation.

"What if the parents are cool but the child is difficult to handle" "Should I watch over them while they play all of the time" "How about if he/she is too energetic jumps up and down and hurts himself/herself?"

and so on....

On one occasion my daughter was invited to a friends house, I took her over, I was invited in for a coffee and spent 15 minutes getting to know each other with the other parent. It was comfortable and reassuring mutually. I liked this approach, and will use it too when possible.

If I had doubts about a family/parents, I would spend a few minutes on foot making polite chit chat, because yes I think that sometimes appearances and perceptions can be misleading.

However if I still had doubts I would either limit the time my child would spend there to a minimum, and of course ultimately you retain the right to cancel or not send your child if you feel like you shouldn't.

The easiest way to deal with this one is to volunteer in your childs classroom. Every primary teacher (grades 1 to 3, here) I've ever known will give their eye-teeth for help in their class. Be there for the class parties, I know it's hard, I had to take time from work when my older kids were younger and used up a lot of my sick days to help with various functions. It's worth it though, you know their friends, and you get to know their parents in what is actually a pretty high-stress environment.

A couple of times I have had mothers stay with their child(ren) for a play date. I assumed at the time that the child(ren) was/were anxious as they didn't know us well, but maybe it was the parents who were anxious. It didn't offend me, so I think it's a valid approach. It probably depends on the age of the child though - these kids were under six.