Confusing quartz crystal impedance graphs
Look at this graph: -

The vertical axis is purely impedance and the parallel impedance coincides with the peak impedance. Adding more parallel capacitance lowers that high-impedance point.
Look at the X-scale - everything happens over a very small frequency range. Some graphs seen on the internet are downright misleading because they don't tell you that what they show is just the X-axis in a small range of a few hertz (for some real crystals).
The parallel resonance is the peak of the magnitude BUT it doesn't have a defined impedance angle (unlike series resonance). Series resonance occurs when \$L_M\$ and \$C_M\$ are nearly cancelling their impedances and we are left with \$R_M\$ in parallel with \$C_P\$ and this is close to zero degrees. I say &quot;nearly cancelling&quot; because to get true 0 degrees phase shift in the impedance there needs to be a slight mismatch.
So, moving on to your 2nd diagram (the one that is possibly seeming to contradict things), there is the anti-resonance point and this corresponds with the phase angle being purely resistive but, in fact it is very high in magnitude. The second diagram only tells you whether the reactance is capacitive or inductive and, possibly misleadingly, gives the impression that overall impedance is comparable with the series resonance case. Not true. Somewhere very slightly displaced from the anti-resonance point is the parallel resonance - it is a peak in magnitude but not at zero degrees.
So, if you were using the crystal as a parallel filter you might naturally choose the parallel resonance point because you wouldn't care too much about the phase angle. This would be the case of the right-hand circuit at the bottom of your question. It is a Colpitts oscillator and the crystal would kill oscillations at anything other than close to parallel resonance.
Here's a simulation using the values in the above drawing. A current source of 1 amp was used to excite the model and the frequency range is from 10.27 MHz to 10.3 MHz: -

And here's a close-up the the parallel/anti resonance area: -

I've positioned left and right cursors as follows: -

Left corresponds with zero degrees phase angle (-0.005748 degrees is as near as I can manipulate this)
Right is positioned to correspond with the impedance maximum peak of 191.285 kohm

Of importance is the delta measurements displayed (inside blue boxes) - they accurately calculate the difference in frequency between left and right cursors and show the delta to be 10.809 degrees i.e. anti-resonance and parallel resonance are displaced by about 11 Hz.