Why do some airports park the aircraft backwards at the gate?
The airports that feature this parking orientation are of small size, with the apron (or part of it) very close to the runway.
Taking Dortmund Airport as an example:

When an aircraft of a certain size (for example Airbus A319, Boeing 737) reaches one of the parking positions 5-12, it is pushed backwards into its parking position. The engines are switched off. Jets of this size, which are parked forward on their own, would cross the obstacle surfaces, in this case the lateral transition surface, with their stabilizer. This, in turn, would result in an increase in obstacle clearance OCA / H (obstacle clearance altitude / height).
Source: de.wikipedia

In other words, the tall tail fins of certain aircraft when parked nose in would violate the runway's lateral transition surface: an imaginary box around the runway that is meant to be clear of high obstructions, to ensure the safe operation of the runway and any navigational equipment.
For Dortmund Airport, I suspect the aircraft are also &quot;pulled-out&quot; before their engine-start due to the lack of jet blast deflectors (the building, personnel, and vehicles, would be unprotected). For example, the blast from the breakaway (taxi) power of an A320 with CFM56 engines is 20 m/s (72 km/h wind) up to ~60 m behind the plane.
It is also why London City has &quot;caution your blast&quot; holding point signs, so aircraft taxiing onto the runway wouldn't use too much thrust with the apron and its personnel and vehicles close behind.

London City Airport showing tails farthest from the close runway, and highlighted in blue is a jet blast deflector behind one of the parking stands. In this airport the pilots steer into this position without the help of a pushback tug.

Example of how the runway area in Dortmund Airport may affect one apron and not the other (yellow circle).