My friend and I parted ____ the door ____ the cabin that I had rented ____ the night”.

The correct option is at, of, for.

To determine the correct set of prepositions, we can analyze each blank in the sentence step-by-step:
1. First blank ("parted ____ the door"): The verb "parted" here means to say goodbye or separate. When referring to a specific physical location where the parting takes place, the preposition "at" is used (e.g., "parted at the door" or "parted at the station"). Thus, "at" is the correct fit for the first blank.
2. Second blank ("the door ____ the cabin"): This describes a relationship of belonging or association between the door and the cabin. The preposition "of" is standard for indicating that the door is a part of or associated with the cabin ("door of the cabin"). Thus, "of" fits the second blank.
3. Third blank ("rented ____ the night"): When referring to a duration of time or the purpose/period for which something is hired or reserved, the preposition "for" is appropriate (e.g., "rented for the night" or "booked for a week"). Thus, "for" is the correct fit for the third blank.

Combining these, we get: "My friend and I parted at the door of the cabin that I had rented for the night". This matches the option "at, of, for".