Series or parallel tank circuit?
Strictly speaking a tank circuit comprises parallel LC components and not series components. Anyway that is what I was taught but that's besides the point.

When you add a capacitor in parallel with the secondary of a transformer you are more than likely (intentionally or not) producing a resonance with the secondary leakage inductance and, given that the secondary leakage inductance is in series with the induced secondary voltage, you are making a 2nd order low pass filter: -




  the voltage rises on it when you drive it at resonance using the
  primary coil


Yes, this can certainly happen with a 2nd order low pass filter formed from leakage inductance and an added capacitor: -



Also take note that the primary leakage inductance is "lumped together" with the secondary leakage inductance when calculating the total effective leakage inductance and it is this total leakage inductance that should be used when calculating the resonant frequency seen at the secondary.