I think this is a simple matter. This is not about judgment or parental rights. It's about business ethics and professional judgment. There will always be parents who will agree to any service for their child, (oddly enough, those are the same folks who will sue you if something goes wrong), that doesn't mean that you have to acquiesce to it. If you say you are a professional, then act like it and exercise some professional judgment. Sure, they might go elsewhere, but at least your conscience is clear. Pretty painted gel nails are not one of life's necessities and not worth the risk to a child. The salons I work with have a strict policy of not using acrylic or gel products on children.
In Salon Geek, we stridently insist that we follow manufacturer's instructions on matter of using the entire system or the right lamp, but then call the manufacturers "overprotective" when it comes to following their instructions about not using their products on children? We disregard their consistent and clear instructions in this regard? That makes no sense.
We know that there is significant risk of sensitivity in adults with acrylic and gel products, even properly used. And while the products may not be dangerous to adults, we know that children's immune and nervous systems are less developed and what is harmless to an adult (anything from honey to small amounts of lead, for example) is harmful to children. Not to mention the potential for long-term exposure to the products, and additional exposure via ingestion because children are always putting their hands in their mouths. Maintenance and aftercare is another difficult issue with children; half our adult clients fail to maintain their nails properly. Do you really expect more of a child?
Bottom line, when it comes to matters of my nails, I trust my nail tech who is a trained professional in her field. But when it come to matters of health and safety of children, I take my guidance from pediatricians and dermatologists - the trained professionals in their field. These professionals are in general agreement each other and with the manufacturers that gel and acrylic products ought not to be used on children under 14.