I'm guessing David Ellis was saddened by the fact that Promethean units didn't stick to their clever chess motif so he kept it prominent in Spartan Ops. Granted, I can't make exact parallels, but:
Lasky is the King of one side. He doesn't come into play until end-game and stays pretty static up until the end game. He also very clearly wants to defend his notable pawn Halsey.
Crimson Squad is a Knight, taking unconventional but repetitive paths to take out stragglers and occasionally bigger targets.
Palmer is a Rook: a heavy unit that doesn't really shine until later on into the match because it can be used as a fearful mindgame and doesn't mind the loss of a mere Pawn.
Jul ''Mdama is also, surprisingly, a Pawn. He faced Halsey on numerous occasions, representing the opening of the match. On the proverbial board, they ended up on squares right next to one another, and both have a similar goal - getting to the end of either side's board to revive a stronger piece. Note that two pawns face-to-face result in a stalemate and cannot "jump" one another; Halsey's apparent "plot armor" isn't broken by Jul or even his Elites, but instead by Palmer.
Halsey got to the end of the board first and, Season One's ending permitting, will likely become a Queen.
Jul looks to do the same, trying to "'make these gifts on our own" (rebuilding his army) or arguably reviving his King, the Didact.
The two players on either side represent the Librarian and arguably humanity or ONI.