The reason is that the parking brake works on the Yellow hydraulic system and that system is powered by an engine driven pump on engine number 2.
It’s different on the A330 and A340, where the parking brake is powered by the Blue hydraulic system. That system’s engine driven pump is on the left hand side. So you start engine 1 (or 1+2 on the A340) first on that type.
That guarantees that you don’t have to rely on the accumulator to provide pressure on the brake when pushback is completed but the engines aren’t started yet or are in the process of being started.
The A320 family, just like the A330s, have 3 hydraulic systems: Green, with a pump on engine 1, Blue, with an electric pump, and Yellow, with a pump on engine 2. On the 330 the engines have 2 hydraulic pumps each: engine 1 the Blue and Green, engine 2 a Yellow and another Green.
(Note how the Yellow hydraulic system of the A320s has the alternate braking system plus parking brake on it. The Green system has the normal braking on it but not the parking brake. Don’t be fooled by the names: the hydraulic fluid isn’t really that colour and the pumps aren’t painted accordingly, it’s just a name thing.)
There’s an accumulator on the Yellow system on the A320 and on the Blue system on the A330. You can start the other engine first but every time you apply the brake on the accumulator the pressure drops in it. So in order not to rely on this, it makes more sense to start the engine that pressurizes the parking brake and accumulator first.
This is an Airbus recommended way of operating but the airlines can modify this. Or if in a certain situation it makes more sense to start the other side first, that sometimes happens too.
(To compare this with the hydraulic architecture of the A330: the Blue system has the brake accumulator and parking brake on it. The blue system of the A330 has an engine driven pump on engine 1 so when that engine is started full hydraulic pressure is available to set the parking brake over and over again, without using up its accumulated pressure.)
