This dad is a coward. Either come out and say what you need to say or shut up with the negative remarks. Kids can sense the hostility.
I had a dcb who would cry when his mom picked him up because he didn't like his play interrupted. Lots of kids do this.
Obviously dad has issues elsewhere in his life and chooses you to bully with his remarks to make himself feel better. He is a immature person.
You being termed over the yogurt cup incident (if you can even call it that!) was not provider misconduct. I refuse (I used to) to take parental issues on my shoulders otherwise I would be up all night thinking about them.
We have a tough job dealing with all these parental personalities. It is like a juggling act sometimes trying to satisfy everyone. If you are happyand confident with your program then don't let them get you down.
If dcd said something negative, ignore it, if he says something else then I would just say in my talking to a toddler voice "you don't sound very happy, is there anything I can help you with?" and smile. Grumpy people are sometimes looking for altercations and they hate it when you are pleasant and proffessional back to them.
Good luck dealing with this guy.