Do older apples contain more or less pectin in the skins/cores?
Older apples should have less pectin, probably:


  The chemistry of pectins is a vast and complex subject ... all these enzymes work together to degrade the pectin and eliminate is sticky effect. They exist naturally in plant tissues ... after harvest, during apple storage, the protopectins within cell walls are naturally broken down ... During that process, the apples get softer ... as the apples become overripe and mealy, the soluble pectins are degraded into nonpectic substances.


Quoted from "The New Cider Maker's Handbook" chapter 12 "The Pectic Substances" by Claude Jolicoeur. (There's also apparently an entire book on this called "The Pectic Substances" by Z. I. Kertesz.)