Minimum current at parallel resonance
Does parallel resonance in an AC circuit always result in maximum
  impedance thus minimum current?


Only when the frequency of the applied signal is correct. A parallel inductor and capacitor have an impedance thus (product over sum): -

\$Z_p = \dfrac{j\omega L}{1 - \omega^2LC}\$

This impedance rises to infinity when the denominator is zero and this happens when: -

\$\omega^2LC = 1\$ or \$\omega = \dfrac{1}{\sqrt{LC}}\$

Or, more conventionally F = \$\dfrac{1}{2\pi\sqrt{LC}}\$

Current drawn from the signal driving the parallel resonant circuit can theoretically acquire a zero value but, there are still voltages present across the individual L and C components and, this results in a large circulating current swishing back and forth between the inductor and capacitor. But, theoretical impedance seen by the driving source can be very high.


  I've read this only happens when the parallel branch 'conductances are
  constant' but have no idea what this means


I also have no idea what this means.