When you say “they eat that stuff up” it’s a colloquial phrase. The context is you say or give something to someone like a praise or gift and the person you are talking to enjoys it so much that they will want more and consume it like a food. Like i tell you, you are pretty and a good person lots of compliments and it flatters you and you don’t say anything back. I could say you’re eating it up (the compliments). In this case he thinks the his gift is something his parents really enjoys and just automatically accepts and loves it without any hesitation or thought.

It means: "but parents love that." It's supposed to be funny because the writer was complaining a few lines back about receiving the same gift for the last four years, but she thinks parents love that. Also, her parents' facial expressions are clearly less than thrilled in the drawing at the bottom.