There is a top level 'stats' section that looks something like this:
replay['stats']['0']['rank'] shows that the player with player index 0 placed 2nd.
As an aside, the engine actually keeps track of a number of other interesting stats for each player but unfortunately only rank gets saved to the replay. The other stats tracked are: last_frame_alive, init_response_time, average_frame_response_time, total_ship_count and damage_dealt.
Edit: Created a pull request to add the missing stats to the replay as PR #296