So the gist of it is that you build an RLC circuit and replace the inductor with this so-called "negative inductor" like so.

And instead of getting a damped oscillation,

You get a parametric oscillation.

And all it costs you is the energy required to maintain the high voltage field between the two plates.

Sounds a little too easy to me.
What I'm guessing in actual practice is the "negative inductor" is actually non-inductive, so attempting to build an RLC circuit with it is going to be rather impossible. What I don't see in the main link is a formula that calculates what the negative inductance is going to be based on the voltage of the electrostatic field applied. Such a formula would be pretty useful and easy to validate its accuracy. To me, a "negative inductor" is really just a capacitor, which makes a RCC circuit seem a little pointless. However, this does give us a means to alter the capacitance by changing the field strength and not the physical characteristics of the capacitor. Essentially a voltage controlled variable capacitor.
What I'd like to know is if the non-inductor can be made just slightly inductive when there is no applied voltage field, then when a field is applied, this non-inductor becomes a capacitor. Such a configuration would be really useful if you can actually change an inductor into a capacitor, back-n-forth just by altering the field intensity. Should be super simple to make a parametric oscillator with such a device. There's some pretty strong evidence the Ruslan device I've been working on, does exactly that via the grenade coil. The Tesla coil flips the grenade from being a coil into a capacitor at just the correct timing. One has to also think about this a little deeper. If two coils can operate as a transformer, what happens when they are switched over to capacitors? Seems to me any back EMF could be completely eliminated if the switching is done at exactly the correct moment.
BTW, does anyone see a non inductive coil between two tubular plates looking a lot like a water capacitor in Stan's WFC? The real question is how do we apply a high voltage field to this non-inductive coil without adding more plates? Suppose the water bath itself generates an electric field...
Heck, maybe it would be fun to wind a caduceus coil around the inner tube (plate) and see what happens for grins and giggles. If I remember correctly, W. B. Smith did an experiment similar to this and discovered he could pour a kilowatt into the darn thing and nothing came out--nothing he could measure anyway.