Well, parallel tubes do increase capacitance on the plates, and they will also draw twice as much current (DC and AC), but as I understand your problem, it's a DC voltage problem. Since the load coil presumably has negligible DC resistance, the DC plate voltage on each tube is the same as before. Now you have two tubes that may overheat, or arc, or whatever.
I think cascode might work, although you might simply use a resistor between B+ and the load coil, with a bypass capacitor to ground from the junction of the resistor and the coil.
Is this an RF circuit? If so, what is the carrier frequency?
BTW, I believe the interelectrode capacitance is primarily from plate to grid. The grid shields the plate from the cathode.