I would definitely talk to her about this and clear the air but maybe this isn't about her not trusting you. Maybe it hasn't got anything to do with anything you are, or are not doing. Maybe she needs to do this to calm her own anxieties. Really if this works for her or she needs a little bit of reassurance I wouldn't necessarily be so quick to jump to the impression that this is about you. Obviously without the facts you can only fill in the blanks and her actions have rubbed you up the wrong way a little but there are far worse things she could be doing. She may not have stopped to ask because she feels like a fool and embarrassed. I wouldn't overthink it to much. Discuss it and maybe encourage her to call if she has any concerns or if she is driving by to feel free to stop if its helpful, otherwise please just know that I have your child's wellbeing and safety as my top priority. Obviously stating the issues with regular drop in that can be disruptive or don't encourage the drop ins if that is against your policy but it does sound to me that this parent just needs some kind words of reassurance.

I don't think this is grounds for termination personally, and I don't think it is a reflection on your capabilities as a daycare provider, although I do understand why you would be made to feel uneasy by this situation.