Why does it suffice to put the audio input in parallel to the battery, in order to add this signal (baseband signal) to the carrier frequency?
Why do we need the capacitor?
This is a very crappy "AM transmitter" circuit which isn't worth its name.
It "works" by modulating the supply voltage with an Audio signal.
The output of a crystal oscillator switches between negative and positive supply. By changing that supply with a signal, the amplitude of the oscillator's output signal will change with that signal.
The "modulation depth" of this circuit will be bad as the crystal will only work for a certain voltage range limiting the amount of AM modulation possible.
It would make more sense to have the 470 ohm resistor in the Bat +5V line instead of the Bat - line as then ground of the audio circuit and this circuit would be the same.
The capacitor is needed to prevent +5 V getting into the audio circuit, without the capacitor the audio circuit might be damaged if it does not have a DC-blocking capacitor of its own.
As I said, this is a really crappy circuit and although it will do "something" it is more a "hack" than a well designed circuit. So don't expect too much from it.