Bughouse?
Bughouse fetch partner game using chess.com API
Yes bughouse, the variant where there are two boards and you have a partner. Whenever you capture pieces on your board, your partner gets them and vice versa. Unfortunately with the way chess.com's analysis boards are laid out, you can't see all of the pieces you have on the side from your partner's captures at a given moment of time. Knowing what you have available to drop is very important to decide what move to play, so getting this info from the API would be helpful. Let me know if it is unclear.
Just wanted to follow-up on this. Would it be at least possible to download the pgn file for each individual game (without the API)? It seems like the download arrow button does nothing for bughouse (or even crazyhouse games). I don't mind then manually/programmatically merging the two pgn files together to form an appropriate .bpgn file, as described in this specification.

Any updates @anara32? I'm also interested in both fetching partner games and BPGN, and noticed the same bug for downloading the games. Is web scraping the best solution?

In the info tab (rather than the moves tab) for the game in live chess there is a link to the partner game. But yeah, this all would be much better in the API.
@nathanf314 AFAIK, web scraping is still the only option. It is extremely slow, but I have made a script for it that works with selenium and scrapes the contents of the movelist on the side of the screen. Loading an entire browser though just to get the games seems non-optimal though, to say the least.
So I am interested in processing my bughouse games from chess.com. Is there a way with the chess.com REST API to fetch the partner's game? I see there are timestamps available, which would be helpful to see the available pieces to drop at any given time.