As mentioned by @pestis above, Parallel Pete is very good with Makeshift Trap. With the updates of Improved Timer, Simple, and Net, you basically have a trap that costs 1 resources every 3 turns with no enemy can do AoO or movement within it. That means, as long as you moves out from that location at the end of your turn, those enemies are just a bunch of harmless toys to you.
Additionally, since he has access to 0-4 Tactic cards and level 0 Guardian cards, the new combo The Book of War + On the Haunt (3) works perfectly on him. Though Makeshift Trap can't reliably kill the enemy every turn to earn 3 bucks, the most important part here is On the Haunt (3) does not require you to draw another encounter card when there is no enemy in the encounter deck. Combine with First Watch, we can cancel the most dangerous encounters at the beginning of the Mytho phase.
Now the stategy is simple. Once you find your Makeshift Trap and The Book of War, dry the enemies as soon as possible with "Where's the party?", Kicking the Hornet's Nest, and On the Haunt (3). Once no secret on The Book of War is left, drop it with Unconventional Method or Act of Desperation, then use your familiar Survivor recursions to bring it back to the table. For any chance you have some nasty enemies to kill in the encounter deck, draw them out as late as possible, then sweep them off with one Dynamite Blast.
Since Pete has a low , at first glance Scavenging does not seem to be a valid option for him. Fortunately, there is one tool in the pool to address this problem: Hyperphysical Shotcaster. With Telescanner, we can sit at a low-shroud location nearby your trap to get clues from any revealed location with ; Combine Translocator with the Guitar, we can kick multiple enemies into the Makeshift Trap in one single turn, or get yourself away from an elite after a successful evasion. The oversuccess requirement on Scavenging is cancelled out by Empowered Configuration. Finally Hyperphysical Shotcaster self-discards, so we can just apply Scavenging without the need of extra discarding events. Costs are not a problem here thanks to Schoffner's Catalogue and other economy cards in the deck.
Combining all above, now you have an investigator that: protects everyone from any non-elite enemies, never draws any treacheries, earns an extra dollar or heal one horror every turn, and consistantly grabs clues when sitting on top of your puppets!