What compounds exist in the bitter peel of specific fruits? Actually, my original question was: What makes a pith of pomegranate so bitter?
Then I decided to be more generic. You'll hardly find someone that consumes pomegranate and doesn't nag about the bitter taste the white pith has! Additionally, when I was a kid, my parents taught me not to eat the bitter peel of an orange, which I later learned that is basic. So, since the evolution of these kind of fruit trees has unanimously been altogether, the chemical species that cause the bitterness of their fruits' piths can be of the same group.
I have a hunch what I seek is the "family" of phenols, but that is nothing more than a hunch. Answer in detail please.
The reason behind the bitterness of most fruits is due to presence of Tannin in them.

Tannin is a naturally occurring polyphenol found in plants, seeds, bark, wood, leaves and fruit skins. 

Tannin tastes bitter,dry and astringent and you can feel it specifically on the middle of your tongue and the front part of your mouth. Besides the peels of fruit, we can feel bitterness in other food products as well like,

High-Tannin Foods =
 Tea Leaves
Walnuts, Almonds and Nuts with Skins
Dark Chocolate
Cinnamon, Clove and other spices
Pomegranates, Grapes 
Quince
Red Beans

The  bitterness and astringency are caused by tannins binding with proteins in our saliva. The chemical constituent of tannins are: gallic acid , ellagic acid and their derivatives(present in hydrolyzable tannins) , catechin and its derivative (present in condensed tannins) 




As for your specific question about the the pomegranate fruit pith, it contains ellagitannins, pelargonidin, punicalin, punicalagin, anthocyanins, cyanidin, ellagic acid which makes it bitter.