Parents have a moral responsibility towards ALL aspects of their children's lives until those kids reach maturity. This does include a policing of the food they eat. After all, that food will turn into their growing bodies.
This does not mean parents cannot allow their children to eat sweets and ice cream and cakes et cetera. It does mean that they should view these as treats, not as routine, everyday food. You could even use these as a reward for certain duties undertaken.
If you're allowing a child to drink factory-produced soft drinks, and recognized 'take away junk food' just about every day, you're preparing them for a life of troubles health-wise. By 'giving in' to their wails of protest for such food, you maybe are getting over an awkward present moment. But in the long run both you and those kids will be the worse off for it.