What compounds exist in the bitter peel of specific fruits?
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.