Programming a parametric equalizer - preventing clipping
There's no universally acclaimed correct design of equalizers. Just like there are multiple car designs per brand, there are also multiple techniques for achieving desired effects. 

It's true that your channel filters will interact with each other and change the effective resulting gain at each frequency, but mostly pronounced at the adjacent channel peak though.

The amount of interaction depends on your selected DC gains (which are unity for your application) and channel bandwidths and bandwidth gains.

There are recommended practices to prevent distortion levels. Headrooms are included so that peak gains will not create clipping. Or you can allow some...? 

Robert Bristow Johnson is a member here and he's one of the designers of those (cookbook) parametric equalizers that you want to use in your application.