YANBU OP.
I'm probably in the minority, but I don't agree with piercings on young children, and definitely not on babies.
If you attend a piercing studio as an adult, you need to confirm you aren't under the influence of alcohol or drugs, because you'd be classed as unable to give consent.
So when a Mother attends with her baby, I do think it's wrong to give consent on the child's behalf for something they didn't want or choose for their own body.
The argument, "She's 4, and she's pestered to have it done" doesn't really work IMHO.
If I, when aged 15, decided to have sex with my 18yo boyfriend, i would be told that by law am not mature enough to give Consent, because even if I wanted to- I'm not of age to sufficiently understand the implications. He, as the adult, would be expected to safeguard and respect the consent issue, to protect both the other party from choices they aren't deemed able to make, and also protect himself from being arrested! You're more aware of your mind and body in your teens than you are as a child, so if a teen is deemed unable to make conscious sensible choices regarding their body, how on earth is a 4 year old?