Saying your child is shy does not escalate the problem. Making it seem as though being shy is a bad thing, escalates the problem. My daughter is shy, and that's ok. Can't really speak for her, but in certain situations she might be afraid to go to and I wouldn't make her. The first time she went over her friend's house after school (and didn't really know the parents well, although I did) she went no problem. So sometimes she surprises me. However, I've seen her be in a classroom with a teacher for the whole year and yet at the end when she needs to ask the teacher something in front of me, she gets scared.
I agree, that if you really want to schedule something, suggest meeting somewhere with the parent. If she declines this offer, than it may be the mom that just doesn't want to do it. Or I agree, with if its boy & girl meeting, that might be the issue too. My daughter, wouldn't be to keen on meeting a boy for a playgroup.