If the parametric exitation is possible, than for the alternator to work the speed of the drive motor should be controlled of fine tune by having a frequency to voltage converter and or phase comparator to fine tune the movement to the resonance. But for me seems very over complicated design.
The thing is that inductance is the number of turns squared divided by the reluctance, so the smaller it is the greater will be the inductance. However the fields are complicated concept in the sense that nothing says that it really exist. The energy you input in a component is inside it somewhere distributed, If you add a lamination of iron inside the transformer you wont need to do work to do so and you would increase the inductance, like naudin said in the 90's but only in one cycle, and here is the problem. If you make the core to oscillate maybe having some springs in it and to the resonant frequency to the spring resonance, you could actually insert and take out the core, much like a speaker works... I already thought of this but didn't build it, i called the vibration transformer or something like that.
I will try to find out more about this parametric component of the equation..
I'm not sure if actually you will get more energy out, but certainly is not the same thing as was tested before so should be tested further.
Br