Hi M. N,
I live a bit too far from where you live, but I'm good corresponding through email :)
I was "painfully shy" also and people often misunderstood me because of it. However, I found a way out of it, or I should say my first daughter did it for me. She came out the most social kid, she comes up to other kids and introduces herself, myself and now also her little sister :P
Kids find a wonderful way of communicating easily I think. Then I began to take advantage of the situation, as my daughter introduces herself to other kids, I force to do so myself also to their mothers (usually the ones with the kids).
I don't know if this helps at all. But do know that other mothers at the park may feel the same way you do and most of the time it just takes a little to begin to have a conversation :)
Also, for your daughter, have you tried your local library? or bookstore? they usually have story times. she might enjoy it and its a chance for her to meet other kids.
If you want to write ____@____.com is my email.
Hugs,
M.