After reading through this thread multiple times and engaging in various discussions on Pokemon Showdown, I've decided on this list of what we want to and what we should be threatened by.
Some things I'd like everyone to keep in mind is that this isn't an end all be all list. If it is in the best interest of the project and concept we'll be able to work around certain parts of this lists as we need to. I've purposefully excluded mentioning some of our more interesting matchups, such as Crucibelle, to allow us more freedom during later stages. There are a lot of more or less neutral matchups that are dependent on variables that have not been decided yet, and as such I will leave it to us to figure out during the natural progression of this project.
Pokemon CAP should threaten
Dark-types: Colossoil, Bisharp, Mega Sableye, etc.
Dragon-types: Cyclohm, Garchomp, etc.
Fighting-types: Tomohawk, Keldeo, etc.
Rock-types: Tyranitar, Stratagem, etc.
These Pokemon are both weak to CAP's STAB moves and have a STAB move resisted by CAP as well. As such they are our natural switch-ins and should be what we threaten directly based off our STAB moves alone. Pokemon that have a secondary typing that can hit CAP 22 super effectively, such as Tomohawk, Latios, and Latias, should be at least reluctant to switch into us or threatened out if CAP 22 can get in safely.
Steel-types, primarily those weak to Fighting-type moves: Heatran, Ferrothorn, etc.
Steel-types are more of a neutral matchup for us as they can threaten us with their STAB moves as much as we can with ours, but we should be able to force out slower Steel-types and make them reluctant to switch into us.
Passive Pokemon: Chansey, Hippowdon, Skarmoy, etc.
Passive Pokemon that aren't directly threatened by our STAB combination may potentially challenge CAP 22 due to their reliance on chip damage over their offensive abilities, which makes them negligibly affected by Parting Shot. Some way to pressure these Pokemon would help CAP 22 in scenarios where these Pokemon don't switch into it directly.
Pokemon that should check/counter CAP:
Poison-types: Mollux, Plasmanta, Fidgit, Mega Venusaur, Amoonguss, etc.
Fire-types: Talonflame, Mega Charizard Y, Volkraken, Volcanion, etc.
Both of the types listed above have the ability to switch into one or more of the CAP's STAB combination and threaten it back with their STAB moves. Both types generally rely on their offensive abilities to deter switch ins and are therefore good targets for Parting Shot.
Priority users: Scizor, Mega Metagross, Azumarill, and Talonflame
These Pokemon are either resistant or are hit neutrally by the CAP's STAB combination and can threaten back with powerful priority before CAP can Parting Shot out. They should be able to force us out in a 1v1 scenario, but whether they should be able to switch in safely is something I'll leave for us to decide during later stages.
In general our goal should be to threaten a large number of Pokemon with our STAB moves in order to force the switches needed to make Parting Shot work, and we should try to confine our checks and counters to ensure that they are affected negatively by Parting Shot.