Haven't had the chance to play Parallel Pete yet but... This is a pretty toothless weakness for the most part, right? Unless you're at a point where you have no cards in hand this doesn't seem that bad, on top of that there are so many survivor cards that WANT to be discarded you could almost turn this into a positive.

I'm thinking Moonstone, the Improvised set of cards, Glimmer of Hope, all manner of cards that actively want out of your hand that aren't always easy to get there. Even if all you ever draw is your 1 card from upkeep this doesn't seem that bad at all. That's not even mentioning the numerous ways survivors can get things out of their discard that you might want to be running anyways! If it was at random it might be scarier.

Overall I'd rate this as a slight annoyance at worst, among the tamest weaknesses in the game. Probably not as meaningless as Amandas or Sister Marys, but still pretty mild. The most destructive part of the weakness is that in order to get it in your deck you have to forgo Duke, and while the Guitar is an extremely powerful (and very unique!) card that's a hard sell.